
For all \(t\in , \) (u(t,), B(t,))\( are divergence free in the sense of distributions and have zero spatial mean.\item Equation (\ref {equa-gMHD}) holds in the sense of distributions, i.e.,for any divergence-free test functions \) C0 (T3)\(,{\begin{@align*}{1}{-1}\int _{\mathbb {T}^3} \partial _t \varphi \cdot u - \nu _1(-\Delta )^{\alpha _1}\varphi \cdot u + \nabla \varphi :( u \otimes u - B \otimes B) {\rm d}x &= 0,\\\int _{\mathbb {T}^3} \partial _t \varphi \cdot B - \nu _2 (-\Delta )^{\alpha _2}\varphi \cdot B + \nabla \varphi :(B \otimes u- u \otimes B) {\rm d}x &= 0.\end{@align*}}\) 
Here, for any \(3\times 3\)  matrices \(A=({A_{ij}})\)  and \(S=({S_{ij}})\) ,
\( A:S=\sum _{i,j=1}^{3}A_{ij}S_{ij}\) .
For simplicity, we consider the solutions on the temporal interval \(\mathbb {T}\) 
to be consistent with the spatial torus \(\mathbb {T}^3\) .
The arguments of this paper can be also extended to any bounded temporal intervals.
The first main result of this paper is formulated below.
Theorem 1.2 
Let \(\alpha _1,\alpha _2\in [0,5/4)\) ,
\((\tilde{u},\tilde{B})\)  be any smooth, divergence-free and mean-free vector fields on 3 ,
and \((s,p,\gamma )\in \mathcal {A}\)  where
\( \mathcal {A}:=\left\lbrace  (s,p,\gamma )\in [0,\frac{3}{2})\times [1, +{\infty })\times [1,+{\infty }): 0\le s< \frac{2}{p}+\frac{1}{\gamma }-\frac{3}{2} \right\rbrace .\) 

(See Figure 1. for the admissible set \(\mathcal {A}\)  of regularities when \(s=0\) .)
Then,
there exists \(\beta ^{\prime }\in (0,1)\) ,
such that for any given \(\varepsilon _*>0\) , there exist a velocity field \(u\)  and a magnetic field \(B\)  such that the following holds:

Weak solution: \((u,B)\)  is a weak solution
to (REF ) in the sense of Definition REF  with zero spatial mean.

Regularity: \(u, B \in H^{\beta ^{\prime }}_{t,x} \cap L^\gamma _tW^{s,p}_x\) .

Temporal support: \(\mathrm {supp\,}_t u \cup \mathrm {supp\,}_t B \subseteq N_{\varepsilon _*}(\mathrm {supp\,}_t \tilde{u}\cup \mathrm {supp\,}_t \tilde{B})\) .

Small deviations on average: \(\Vert u-\tilde{u}\Vert _{L^1_tL^2_x}\le \varepsilon _*\) , \(\Vert B-\tilde{B}\Vert _{L^1_tL^2_x}\le \varepsilon _*\) .

Small deviations of magnetic helicity: \( \Vert \mathcal {H}_{B,B}-\mathcal {H}_{\widetilde{B},\widetilde{B} }\Vert _{L_t^1}\le \varepsilon _* \) .


<FIGURE>As a consequence, Theorem REF 
gives the following result concerning the non-uniqueness of weak solutions
and the non-conservation of magnetic helicity.
Corollary 1.3 
For any \({\alpha }_1,{\alpha }_2\in [0,5/4)\)  and
any \((s,p,\gamma )\in \mathcal {A}\) ,
where the regularity admissible set \(\mathcal {A}\)  is as in (REF ),
there exist infinitely many weak solutions to (REF )
with the same data at time zero,
which live in the space \(H^{\beta ^{\prime }}_{t,x} \cap L^\gamma _tW^{s,p}_x \) 
for some \(\beta ^{\prime }>0\) 
and do not conserve the magnetic helicity.

Remark 1.4
(i). The non-uniqueness results for the ideal MHD (REF )
were proved by Faraco-Lindberg-Székelyhidi [1]}
for weak solutions with compact support in space-time,
and by Beekie-Buckmaster-Vicol [2]} for
weak solutions with non-trivial magnetic helicity.
To the best of our knowledge, Corollary REF  provides
the first non-uniqueness result for the weak solutions to generalized MHD equations (REF ) with \(\alpha _i\in (0,5/4)\) , \(i=1,2\) .
The value \(5/4\)  actually coincides with the Lions exponent for the well-posedness of the hyperviscous NSE [3]},
and with the exponent \(1/2 + n/4\)  for the well-posedness of 3D generalized MHD [4]}.
(ii). By the Onsager-type conjecture for the magnetic helicity in [5]},
it is expected that \(L^3_{t,x}\)  is the threshold for the magnetic helicity conservation for the ideal MHD. In view of Corollary REF ,
we see that the magnetic helicity conservation fails for the weak solutions in
the space \(H^{\beta ^{\prime }}_{t,x} \cap L^\gamma _tW^{s,p}_x\) ,
where \(\beta ^{\prime }\)  is a small positive constant and \((s,p,\gamma ) \in \mathcal {A}\)  with \(\mathcal {A}\)  as in Theorem REF .
In particular, Corollary REF  provides more examples
in the spaces \(L^3_tL^{{12}/{7}}_x\)  and \(L^{6/5}_tL^3_x\) 
for the flexible part of the conjecture in [5]}.
It would be very interesting to fill the spatial gap from \(L^{12/7}_x\)  to \(L^3_x\) 
or the temporal gap from \(L^{6/5}_t\)  to \(L^3_t\) .
(iii).
In the higher dimensions \(d\ge 4\) ,
one may prove the non-uniqueness of weak solutions in \(C_tL^2_x\) 
for the viscous and resistive MHD (i.e., \({\alpha }_i=1\)  in (REF ), \(i=1,2\) ).
The solutions can be continuous in time.
This is possible because, in high dimensions \(d\ge 4\) ,
there is more freedom to choose the oscillation directions in the
velocity and magnetic flows,
which permits to gain the 3D spatial intermittency
to control the Laplacian \((-\Delta )\) .
(iv).
We expect that the proof of this paper applies to the non-uniqueness in law for the stochastic generalized MHD equations
driven by Wiener processes,
which will be done in a forthcoming work.
We would like to refer to [7]}, [8]}, [9]}, [10]} for the recent developments in the stochastic contexts.

Our next result is concerned with the
vanishing viscosity and resistivity limits of the weak solutions to
generalized MHD equations.
Theorem 1.5 
Let \({\alpha }_1,{\alpha }_2\in (0,5/4)\)  and
\((u,B)\in H^{\widetilde{\beta }}_{t,x} \times H^{\widetilde{\beta }}_{t,x}\)  be any mean-free weak solution to the ideal MHD (REF ),
where \(\widetilde{\beta }>0\) .
Then, there exist \(\beta ^{\prime } \in (0, \widetilde{\beta })\)  and a sequence of weak solutions
\((u^{(\nu _{n})},B^{(\nu _n)})\in H^{\beta ^{\prime }}_{t,x} \times H^{\beta ^{\prime }}_{t,x}\) 
to the generalized MHD (REF ),
where \(\nu _n=(\nu _{1,n},\nu _{2,n})\) 
and \(\nu _{1,n}\) , \(\nu _{2,n}\) 
are the viscosity and resistivity coefficients, respectively,
such that
as \(\nu _{n}\rightarrow 0\) ,
\(u^{(\nu _{n})}\rightarrow u,\quad B^{(\nu _{n})}\rightarrow B \quad \text{strongly in}\ H^{\beta ^{\prime }}_{t,x}.\) 

Remark 1.6
(i). By Theorem REF , the set of the
accumulation points of weak solutions to the generalized MHD equations,
in the \(H^{\beta ^{\prime }}_{t,x}\)  topology,
contains the weak solutions to the ideal MHD equations.
This extends the vanishing viscosity result in
the context of the NSE [11]}
to the generalized MHD equations.
Let us mention that, the Sobolev space \(H^{\widetilde{\beta }}_{t,x}\)  is less regular than \(C^{\widetilde{\beta }}_{t,x}\) 
used in the vanishing viscosity limit result for the NSE in [11]}.
In order to get the temporal regularity of solutions,
we use the Slobodetskii-type norm of \(H^{\widetilde{\beta }}_{t,x}\)  to gain the regularity on average,
which enables to prove the vanishing viscosity limit in the Sobolev space \(H^{\beta ^{\prime }}_{t,x}\) 
with \(\beta ^{\prime }<\widetilde{\beta }\) . See the proof of () and () below.
(ii). Another interesting outcome of Theorem REF  is
related to the weak solutions constructed in [2]}
and to the Taylor conjecture concerning the conservation of magnetic helicity.
The Taylor conjecture is mathematically formulated in the concept of
weak ideal limits,
i.e., the weak limits of Leray weak solutions,
and has been recently proved by
Faraco-Lindberg [14]}.
The crucial ingredient of the proof
is the regularity of Leray approximating solutions.
Hence,
the weak solutions constructed in [2]} with non-trivial magnetic helicity cannot be obtained
as the weak ideal limits of Leray-Hopf solutions to the MHD
(i.e., (REF ) with \({\alpha }_i=1\) , \(i=1,2\) ).
Let us mention that, the weak solutions constructed in [2]}
has the regularity \(H^{\widetilde{\beta }}_{t,x}\)  for some \(\widetilde{\beta }>0\) 
(see Remark REF  below).
Hence, by virtue of Theorem REF ,
we note that, in contrast to the weak ideal limits,
the weak solutions in [2]} can be obtained
as the vanishing viscosity and resistivity limits
of those to more general MHD equations (REF ),
where the exponents \({\alpha }_i\)  can be even larger than one.
In particular,
this yields that Taloy's conjecture does not hold along the
vanishing viscosity and resistivity limits of the weak solutions to
the (generalized) MHD equations.

Outline of the proof
Our strategy of proof is based on the intermittent convex integration scheme,
inspired by the works [2]}, [11]}, [20]}.
More precisely, for each integer \(q \in \mathbb {N}\) , we consider the following relaxation system
\(\left\lbrace \aligned &\partial _t {u_q}+\nu _1(-\Delta )^{\alpha _1} {u_q}+ {\mathrm {div}}({u_q}\otimes {u_q}-+\nabla P_q={\mathrm {div}}\mathring{R}_q^u, \\&\partial _t \nu _2 (-\Delta )^{\alpha _2}  {\mathrm {div}}({u_q}-{u_q}\otimes ={\mathrm {div}}\mathring{R}_q^B, \\&{\mathrm {div}}{u_q}= 0, \quad {\mathrm {div}} 0,\endaligned \right.\) 
where the Reynolds stress \(\mathring{R}_q^u\)  is a symmetric traceless \(3\times 3\)  matrix,
and the magnetic stress \(\mathring{R}_q^B\)  is a skew-symmetric \(3\times 3\)  matrix.
The quantitative estimates of the relaxation solutions \((u_q, B_q, \mathring{R}^u_q, \mathring{R}^B_q)\) 
are specified in the main iteration result below.
More precisely, let \(\alpha :=\max \lbrace \alpha _1,\alpha _2\rbrace \) ,
\(\alpha _* :=\max \lbrace 2\alpha -1,0\rbrace \) .
Take \(\varepsilon \in \mathbb {Q}_+\) 
sufficiently small such that
\(\varepsilon \le \frac{3-2\alpha _*}{20},\) 
and
\( s+\frac{3}{2} - \frac{2}{p} - \frac{1}{\gamma }+ 8 {\varepsilon }<0.\) 

For \(q\in \mathbb {N}\) ,
the frequency parameter \(\lambda _q\)  and
the amplitude parameter \(\delta _{q+2}\)  are defined by
\(\lambda _q=a^{(b^q)}, \ \ \delta _{q+2}=\lambda _{q+2}^{-2\beta }.\) 
Here, \(a\in 5\mathbb {N}\)  is a large integer of multiple 5 such that
\(a^\varepsilon \in 5\mathbb {N}\) ,
\(b\in 2\mathbb {N}\)  is a large integer of multiple 2
such that \(\lambda _{q+1}^{2\varepsilon }=\lambda _q^{2\varepsilon b}\in \mathbb {N}\) 
and
\(b>\frac{1000}{\varepsilon }, \ \ 0<\beta <\frac{1}{100b^2}. \) 

Let \(q \in \mathbb {N}\) ,
we assume the following inductive estimates of the relexation solutions to (REF ) at level \(q\) :
\(& \Vert {u_q}\Vert _{C_{t,x}^{1}} \le \lambda _{q}^{7},\quad \Vert _{C_{t,x}^{1}} \le \lambda _{q}^{7},  \\& \Vert \mathring{R}_q^u\Vert _{C^{1}_{t,x}} \le \lambda _{q}^{14},\quad \Vert \mathring{R}_q^B\Vert _{C^{1}_{t,x}} \le \lambda _{q}^{14}, \\& \Vert \mathring{R}_q^u\Vert _{L^{1}_{t,x}} \le \delta _{q+1},\quad \Vert \mathring{R}_q^B\Vert _{L^{1}_{t,x}} \le \delta _{q+1}. \) 
The main iteration result is contained in the following theorem.
Theorem 1.7 (Main iteration) 
Let \(\alpha _1,\alpha _2 \in [0,5/4)\) .
Then, there exist \(\beta \in (0,1)\) ,
\(M>0\)  large enough and \(a_0=a_0(\beta , M)\) ,
such that for any integer \(a\ge a_0\) ,
the following holds:
Suppose that \(({u_q},\mathring{R}_q^u,\mathring{R}_q^B)\)  solves (REF )
and satisfies (REF )-().
Then, there exists \((u_{q+1}, B_{q+1}, \mathring{R}^u_{q+1}, \mathring{R}^B_{q+1})\) 
solving (REF ) and satisfying (REF )-() with \(q+1\)  replacing \(q\) .
In addition, we have
\(\left\Vert u_{q+1}-u_{q}\right\Vert _{L^{2}_{t,x}} \le M\delta _{q+1}^{\frac{1}{2}}, \quad \left\Vert B_{q+1}-B_{q}\right\Vert _{L^{2}_{t,x}} \le M\delta _{q+1}^{\frac{1}{2}},\) 
\(\left\Vert u_{q+1}-u_{q}\right\Vert _{L^1_tL^{2}_{x}} \le \delta _{q+2}^{\frac{1}{2}}, \quad \left\Vert B_{q+1}-B_{q}\right\Vert _{L^1_tL^{2}_{x}} \le \delta _{q+2}^{\frac{1}{2}},\) 
and
\(&\mathrm {supp\,}_t u_{q+1} \cup \mathrm {supp\,}_t B_{q+1}\subseteq N_{\delta _{q+2}^{\frac{1}{2}}}\left(\mathrm {supp\,}_t u_{q} \cup \mathrm {supp\,}_t B_{q} \cup \mathrm {supp\,}_t \mathring{R}_{q}^u\cup \mathrm {supp\,}_t \mathring{R}_{q}^B\right) ,\\& \mathrm {supp\,}_t \mathring{R}_{q+1}^u \cup \mathrm {supp\,}_t \mathring{R}_{q+1}^B\subseteq N_{\delta _{q+2}^{\frac{1}{2}}}\left(\mathrm {supp\,}_t u_{q} \cup \mathrm {supp\,}_t B_{q} \cup \mathrm {supp\,}_t \mathring{R}_{q}^u\cup \mathrm {supp\,}_t \mathring{R}_{q}^B\right). \) 

The heart of the proof of Theorem REF  is to
construct suitable perturbations
\(w_{q+1} \simeq u_{q+1} - u_q,\ \ d_{q+1} \simeq B_{q+1}- B_q,\) 

module small mollification errors,
such that
the corresponding nonlinear effects decrease the amplitudes of the Reynolds and magnetic stresses,
in order to fulfill the above iteration procedure.

New ingredients of proof
One major difference between the NSE and the ideal MHD equations is that,
because of the strong coupling of the velocity and magnetic fields,
besides the intermittent velocity flows \(\lbrace W_{(k)}\rbrace \) ,
the intermittent magnetic flows \(\lbrace D_{(k)}\rbrace \)  shall be also introduced
in order to respect the geometry of MHD.
This forces to limit the oscillation directions,
and thus the intermittent velocity and magnetic flows
do not have the 3D spatial intermittency as in the NSE case [11]}.
In the context of the ideal MHD (REF ),
the key intermittent shear velocity flows
and the intermittent shear magnetic flows
were introduced of the form
\( W_{(k)} = \phi _{(k)} k_1,\quad D_{(k)} = \phi _{(k)} k_2,\) 

where \(\lbrace \phi _{(k)}\rbrace \)  are the spatial concentration functions
with one oscillation direction \(k\) , orthogonal to the other two directions \(k_1,k_2\) .
The shear flows have the almost 1D spatial intermittency, i.e.,
\( \Vert W_{(k)}\Vert _{C_tL^1_x} + \Vert D_{(k)}\Vert _{C_tL^1_x}\lesssim \lambda ^{-\frac{1}{2} +{\varepsilon }},\) 

which permits to control the
fractional viscosity and resistivity
\((-\Delta )^{{\alpha }_i}\)  with \({\alpha }_i\in [0,1/2)\) , \(i=1,2\) .
One of the novelties of our proof
is the construction of a new class of intermittent velocity and magnetic flows,
which feature the refined intermittency in both the space and time
and thus enable us to control the stronger viscosity and resistivity
\((-\Delta )^{{\alpha }_i}\)  with \({\alpha }_i\ge 1/2\) , \(i=1,2\) .

(i) Spatial intermittent building blocks.
We construct the refined intermittent
velocity and magnetic flows
\( W_{(k)} = \psi _{(k_1)}\phi _{(k)} k_1,\quad D_{(k)} = \psi _{(k_1)}\phi _{(k)} k_2,\) 
which are \(({/({\lambda r_{\perp }}))^3-periodic andconcentrate on smaller cuboids with volume \lambda ^{-3}r_{\perp }^{-3}r_{\parallel }in each of the cubes with side length {2\pi }/{(\lambda r_{\perp })}.Here,\psi _{(k_1)} and \phi _{(k)}  are suitable concentration functions(see (\ref {snp}) below),r_{\parallel } and r_{\perp } are the small constantsto parameterize the spatial concentration of the building blocks,as in the intermittent jets for the NSE \cite {bv19r}.The refined flows enable to gain the almost 2D-intermittency, i.e.,{\begin{@align}{1}{-1} \Vert W_{(k)}\Vert _{C_tL^1_x} + \Vert D_{(k)}\Vert _{C_tL^1_x}\lesssim \lambda ^{-1 + {\varepsilon }},\end{@align}}and thus enables to control the fractional viscosity and resistivity (-\Delta )^{{\alpha }_i} with {\alpha }_i\in [1/2, 1), i=1,2.}Another nice feature of the refined intermittent velocity and magnetic flows is thatthe spatial supports are mutually disjoint,as the Mikado flows in \cite {ds17},which in particular simplifies the control of the interactions between different flows.\) Let us also mention that,
unlike the one oscillation direction \(k\)  in (REF )
in the ideal MHD case,
the building blocks \(W_{(k)}\)  and \(D_{(k)}\)  in (REF )
oscillate in two orthogonal directions \(k_1\)  and \(k\) .
This new oscillation direction \(k_1\) 
causes extra high spatial oscillations,
which will be balanced
by the introduction of a new temporal parameter \(\mu \) 
in the flows
and the temporal correctors \(w^{(t)}_{q+1}\)  and \(d^{(t)}_{q+1}\) .
See (REF ), ()
and the important indentities (REF ) and (REF ) below.
It might be tempting to introduce one more oscillation direction \(k_2\)  in the building blocks,
i.e., with the oscillation directions \((k,k_1,k_2)\)  as in the NSE context [11]}, [23]}.
However,
this causes even more high spatial oscillations, which seem not to be balanced by
the temporal parameter \(\mu \) .

(ii) Temporal intermittent building blocks.
The next major difficulty is to pass beyond the border line \({\alpha }_i =1\) , \(i=1,2\) .
The key idea here is to exploit the new intermittency from
the temporal oscillations.
Thus, rather than performing the pointwise analysis in time,
we measure the Reynolds and magnetic stresses on average,
i.e., in the space \(L^1_{t,x}\) .
Let us mention that,
this is in spirit close to the works [11]}, [25]}.
In [25]} the temporal intermittency is crucial
to achieve the non-uniqueness of weak solutions for transport equations
at the critical space regularity.
In the NSE context [11]},
the key ingredient is the \(L^2_x\) -based spatial intermittency,
instead of the previous pointwise analysis in space in the Euler settings.
We introduce the temporal concentration function \(g_{(\tau )}\) 
in the principal part of perturbations \((w_{q+1}^{(p)}\) , \(d_{q+1}^{(p)})\) ,
and control the oscillation errors in the space \(L^1_t\) .
Here, \(\tau \)  is the constant to parameterize the temporal concentration.
In particular,
this permits to gain an additional almost 1D intermittency, i.e.,
\(\Vert g_{(\tau )}\Vert _{L^1_t} \lesssim \lambda ^{-\frac{1}{2} + {\varepsilon }}.\) 

Quite interestingly,
the gained temporal intermittency enables to control the viscosity and resistivity \((-\Delta )^{{\alpha }_i}\) 
for all the exponents \({\alpha }_i\)  less than \(5/4\) , \(i=1,2\) ,
that is, exactly the Lions exponents.
Let us also mention that,
extra high temporal oscillations arise due to
the presence of the temporal concentration function \(g_{(\tau )}\) .
In order to balance these high oscillations,
another new type of temporal correctors \(w_{q+1}^{(o)},d_{q+1}^{(o)}\) 
will be introduced, respectively, for the velocity and magnetic flows.
See (REF ), ()
and the important identities (REF ) and (REF ) below.

The remainder of this paper is organized as follows.
We first regularize the relexation solutions to (REF )
by the standard mollification procedure in Section .
Section  is devoted to the construction of the crucial intermittent
velocity and magnetic flows,
which feature the intermittency in both the space and the time.
Then, in Section  we construct the velocity and magnetic perturbations,
consisting of the principal parts to decrease the Reynolds and magnetic stresses,
the incompressibility correctors and two types of temporal correctors.
Several key algebraic identities and analytic estimates will also be given,
which lead to the verification of the inductive estimates for the velocity and magnetic fields.
Then, the inductive estimates of Reynolds and magnetic stresses
are verified in Section .
Consequently, the main results are proved in Section .

Mollification
In order to avoid the loss of derivatives, we mollify the velocity and magnetic fields.
Let \(\phi _{\varepsilon }\)  and \(\varphi _{\varepsilon }\)  be the mollifiers on 3 and \(, respectively, \) >0\(,and \)\(\mathrm {supp\,}\) (-, )\(.\) The mollifications of \(({u_q},\mathring{R}_q^u,\mathring{R}_q^B)\)  in space and time are defined by
\(\begin{array}{ll}& u_{\ell }:=\left(u_{q} *_{x} \phi _{\ell }\right) *_{t} \varphi _{\ell }, \quad B_{\ell }:=\left(_{x} \phi _{\ell }\right) *_{t} \varphi _{\ell }, \\& \mathring{R}_\ell ^u:=\left(\mathring{R}_q^u*_{x} \phi _{\ell }\right) *_{t} \varphi _{\ell },\quad \mathring{R}_\ell ^B:=\left(\mathring{R}_q^B*_{x} \phi _{\ell }\right) *_{t} \varphi _{\ell },\end{array}\) 
where the scale of mollification is parameterized by
\( \ell :=\lambda _q^{-20}.\) 
Then, by equation (REF ),
\((u_{\ell },B_{\ell },\mathring{R}_\ell ^u,\mathring{R}_\ell ^B)\)  satisfies
\(\left\lbrace \aligned &\partial _{t} u_{\ell }+\nu _1(-\Delta )^{\alpha _1} u_{\ell }+{\mathrm {div}}\left(u_{\ell }\otimes u_{\ell }-B_{\ell }\otimes B_{\ell }\right)+\nabla P_{\ell }={\mathrm {div}}(\mathring{R}_{\ell }^{u}+\mathring{R}_{com}^{u}), \\&\partial _{t} B_{\ell }+\nu _{2}(-\Delta )^{\alpha _2} B_{\ell }+{\mathrm {div}}\left(B_{\ell }\otimes u_{\ell }-u_{\ell } \otimes B_{\ell }\right)={\mathrm {div}}(\mathring{R}_{\ell }^{B}+\mathring{R}_{com}^{B}),\\&{\mathrm {div}}u_{\ell }=0, \quad {\mathrm {div}}B_{\ell }=0, \\\endaligned \right.\) 
where the traceless symmetric commutator stress \(\mathring{R}_{c om}^{u}\)  and
the skew-symmetric commutator stress \(\mathring{R}_{c om}^{B}\)  are of form
\(\begin{aligned}&\mathring{R}_{com}^{u} :=u_{\ell }\mathring{\otimes } u_{\ell }-B_{\ell } \mathring{\otimes }B_{\ell }-\left({u_q}\mathring{\otimes }{u_q}-{\otimes }) *_{x} \phi _{\ell } *_{t} \varphi _{\ell },\\&\mathring{R}_{com}^{B} :=B_{\ell } \otimes u_{\ell }-u_{\ell }\otimes B_{\ell }-\left({u_q}-{u_q}\otimes ) *_{x} \phi _{\ell } *_{t} \varphi _{\ell },\right.\right.and the pressure P_{\ell } is given by\begin{equation}\begin{aligned}&P_{\ell } :=\left(P_{q} *_{x} \phi _{\ell }\right) *_{t} \varphi _{\ell }-\left|u_{\ell }\right|^{2}+\left|B_{\ell }\right|^{2}+\left(\left|{u_q}\right|^{2}-\left||^{2}\right) *_{x} \phi _{\ell } *_{t} \varphi _{\ell }.\right.\end{aligned}\end{equation}By the standard mollification estimates and the inductive estimates (\ref {ubc})-(\ref {rub}),for any N\in \mathbb {N}_+,{\begin{@align}{1}{-1}& \left\Vert u_\ell \right\Vert _{C_{t,x}^N}+\left\Vert B_\ell \right\Vert _{C_{t,x}^N}\lesssim \ell ^{-N+1}(\left\Vert u_q\right\Vert _{C_{t,x}^1}+\left\Vert B_q\right\Vert _{C_{t,x}^1}) \lesssim \ell ^{-N+1}\lambda _q^7\lesssim \ell ^{-N},  \\& \Vert  \mathring{R}_\ell ^u \Vert _{C_{t,x}^N} + \Vert \mathring{R}_\ell ^B \Vert _{C_{t,x}^N} \lesssim \ell ^{-N+1} ( \Vert  \mathring{R}_q^u \Vert _{C_{t,x}^1}+ \Vert \mathring{R}_q^B \Vert _{C_{t,x}^1} ) \lesssim \ell ^{-N+1} \lambda _{q}^{14}\lesssim \ell ^{-N},  \\& \Vert  \mathring{R}_\ell ^u \Vert _{L^{1}_{t,x}}\lesssim \delta _{q+1}, \quad \Vert  \mathring{R}_\ell ^B \Vert _{L^{1}_{t,x}} \lesssim \delta _{q+1}. \end{@align}}Moreover,by the inductive estimate (\ref {ubc}) and the double commutator estimate(see, e.g., \cite [Proposition B.1]{luo19}, \cite [Lemma 1]{CD12}, \cite {cwt94}),for 1<p<+{\infty },\begin{equation}\Vert \mathring{R}_{com}^{u} \Vert _{L^{1}_{t}L^p_x} \lesssim \Vert \mathring{R}_{com}^{u} \Vert _{C_{t,x}}\lesssim \ell (\Vert u_q\Vert _{C_{t,x}^1}^2+\Vert B_q\Vert _{C_{t,x}^1}^2 )\lesssim \ell \lambda _{q}^{14},\end{equation}and, similarly,\begin{equation}\Vert \mathring{R}_{com}^{B} \Vert _{L^{1}_{t}L^p_x}\lesssim \Vert \mathring{R}_{com}^{B} \Vert _{C_{t,x}}\lesssim \ell \lambda _{q}^{14}.\end{equation}\end{aligned}\) 

Intermittent velocity and magnetic flows
This section contains the crucial intermittent velocity and magnetic flows,
which are indeed the fundamental building blocks in the convex integration scheme.
We set the parameters \(r_{\perp }\) , \(r_{\parallel }\) , \(\lambda \) , \(\mu \) , \(\tau \)  and \(\sigma \)  as follows
\(r_{\perp }:= \lambda _{q+1}^{-1+2\varepsilon },\ r_{\parallel }:= \lambda _{q+1}^{-1+6\varepsilon },\ \lambda := \lambda _{q+1},\ \mu :=\lambda _{q+1}^{\frac{3}{2}-6\varepsilon }, \ \tau :=\lambda _{q+1}^{1-6\varepsilon }, \ \sigma :=\lambda _{q+1}^{2\varepsilon },\) 
where \(\varepsilon \)  is the small constant satisfying (REF ).
Geometric lemmas.
To begin with, let us first recall two geometric lemmas in [2]}.
Lemma 3.1 (First Geometric Lemma, [2]})

There exists a set \(\Lambda _B \subset S^2 \cap \mathbb {Q}^3\)  that consists of vectors \(k\)  with associated orthonormal bases \((k, k_1, k_2)\) , \(\varepsilon _B > 0\) , and smooth positive functions \(\gamma _{(k)}: B_{\varepsilon _B}(0) \rightarrow \mathbb {R}\) , where \(B_{\varepsilon _B}(0)\)  is the ball of radius \(\varepsilon _B\)  centered at 0 in the space of \(3 \times 3\)  skew-symmetric matrices, such that for \(A \in B_{\varepsilon _B}(0)\)  we have the following identity:
\(A = \sum _{k \in \Lambda _B} \gamma _{(k)}^2(A) (k_2 \otimes k_1 - k_1 \otimes k_2) .\) 

Lemma 3.2 (Second Geometric Lemma, [2]})

There exists a set \(\Lambda _u \subset S^2 \cap \mathbb {Q}^3\)  that consists of vectors \(k\)  with associated orthonormal bases \((k, k_1, k_2)\) , \(\varepsilon _u > 0\) , and smooth positive functions \(\gamma _{(k)}: B_{\varepsilon _u}(\mathrm {Id}) \rightarrow \mathbb {R}\) , where \(B_{\varepsilon _u}(\mathrm {Id})\)  is the ball of radius \(\varepsilon _u\)  centered at the identity in the space of \(3 \times 3\)  symmetric matrices, such that for \(S \in B_{\varepsilon _u}(\mathrm {Id})\)  we have the following identity:
\(S = \sum _{k \in \Lambda _u} \gamma _{(k)}^2(S) k_1 \otimes k_1 .\) 
Furthermore, we may choose \(\Lambda _u\)  such that \(\Lambda _B \cap \Lambda _u = \emptyset \) .

As pointed out in [2]},
there exists \(N_{\Lambda } \in \mathbb {N}\)  such that
\( \lbrace  N_{\Lambda } k,N_{\Lambda }k_1 , N_{\Lambda }k_2 \rbrace  \subset N_{\Lambda } \mathbb {S}^2 \cap \mathbb {Z}^3.\) 
For instance, \(N_{\Lambda } = 65\)  suffices.
We also denote by \(M_*\)  the geometric constant such that
\(\sum _{k \in \Lambda _{u}} \Vert \gamma _{(k)}\Vert _{C^4(B_{\varepsilon _u}(\mathrm {Id}))}+ \sum _{k \in \Lambda _{B}} \Vert \gamma _{(k)}\Vert _{{C^4(B_{\varepsilon _B}}(0))} \le M_*.\) 

This parameter is universal and will be used later in the estimates of the size of perturbations.

Spatial building blocks.
Let \(\Phi : \mathbb {R} \rightarrow \mathbb {R}\)  be a smooth cut-off function supported on
the interval \([-1,1]\) .
We normalize \(\Phi \)  such that \(\phi := - \frac{d^2}{dx^2}\Phi \)  satisfies
\(\frac{1}{2 \pi }\int _{\mathbb {R}} \phi ^2(x){\rm d}x = 1.\) 
Moreover, let \(\psi : \mathbb {R} \rightarrow \mathbb {R}\)  be a smooth and mean zero function,
supported on the interval \([-1,1]\) , satisfying
\(\frac{1}{2 \pi } \int _{\mathbb {R}} \psi ^{2}\left(x\right) {\rm d}x=1.\) 
The corresponding rescaled cut-off functions are defined by
\(\phi _{r_{\perp }}(x) := {r_{\perp }^{-\frac{1}{2}}}\phi \left(\frac{x}{r_{\perp }}\right), \quad \Phi _{r_{\perp }}(x):= {r_{\perp }^{-\frac{1}{2}}} \Phi \left(\frac{x}{r_{\perp }}\right),\quad \psi _{r_{\parallel }}\left(x\right) := {r_{\Vert }^{- \frac{1}{2}}} \psi \left(\frac{x}{r_{\Vert }}\right).\) 
Note that, \(\phi _{r_{\perp }}=-r_{\perp }^2 \frac{d^2}{dx^2} \Phi _{r_{\perp }}\) ,
and \(\phi _{r_{\perp }}, \psi _{r_{\parallel }}\)  are supported in the ball of radius \(r_{\perp }\)  and \(r_{\parallel }\) , respectively, in \({\mathbb {R}}\) .
By an abuse of notation,
we periodize \(\phi _{r_{\perp }}\) , \(\Phi _{r_{\perp }}\)  and \(\psi _{r_{\parallel }}\)  so that
they are treated as periodic functions defined on \(\mathbb {T}\) .

The intermittent velocity flows are defined by
\(W_{(k)} := \psi _{r_{\parallel }}(\lambda r_{\perp }N_{\Lambda }(k_1\cdot (x-{\alpha }_{(k)})+\mu t))\phi _{r_{\perp }}( \lambda r_{\perp }N_{\Lambda }k\cdot (x-{\alpha }_{(k)}))k_1,\ \ k \in \Lambda _u \cup \Lambda _B ,\) 
and the intermittent magnetic flows are defined by
\(D_{(k)} := \psi _{r_{\parallel }}(\lambda r_{\perp }N_{\Lambda }(k_1\cdot (x-{\alpha }_{(k)})+\mu t))\phi _{r_{\perp }}( \lambda r_{\perp }N_{\Lambda }k\cdot (x-{\alpha }_{(k)}))k_2, \ \ k \in \Lambda _B .\) 
Here, \(N_{\Lambda }\)  is given by (REF ) above,
\((k,k_1,k_2)\)  are the orthonormal bases in \({\mathbb {R}}^3\)  in Lemmas REF  and REF .
The parameters \(r_{\parallel }\)  and \(r_{\perp }\)  parameterize the concentration of the flows,
and \(\mu \)  is the temporal oscillation parameter.
Moreover, the vectors \({\alpha }_{(k)} \in {\mathbb {R}}^3\) , \(k\in \Lambda _u\cup \Lambda _B\) , are shifts to ensure that the intermittent flows
\(\lbrace W_{(k)}, D_{(k)}\rbrace \)  have mutually disjoint supports.
The existence of vectors \({\alpha }_{(k)}\) 
is analogous to that of Mikado flows in [23]}, [33]}.
In fact, \(\lbrace W_{(k)}, D_{(k)}\rbrace \)  are \((\mathbb {T}/(\lambda r_{\perp }))^3\) -periodic,
supported on the cuboids with length \(\sim {1}/{\lambda r_{\perp }}\) ,
width \(\sim {r_{\parallel }}/{(\lambda r_{\perp })}\)  and height \(\sim {1}/{\lambda }\) .
Because the width and height are much smaller than the length,
each cuboid can be viewed as a straight line.
As there are only a finite number of wavevectors in \(\Lambda _B\)  and \(\Lambda _u\) ,
one may choose the shifts \(\lbrace {\alpha }_{(k)}\rbrace \)  such that
these straight lines do not intersect.
(See the intermittent flows in Figure 2.)
<FIGURE>To ease notations, we set
\(\begin{array}{ll}&\phi _{(k)}(x) := \phi _{r_{\perp }}( \lambda r_{\perp }N_{\Lambda }k\cdot (x-{\alpha }_{(k)})), \ \ \Phi _{(k)}(x) := \Phi _{r_{\perp }}( \lambda r_{\perp }N_{\Lambda }k\cdot (x-{\alpha }_{(k)})), \\&\psi _{(k_1)}(x) :=\psi _{r_{\parallel }}(\lambda r_{\perp }N_{\Lambda }(k_1\cdot (x-{\alpha }_{(k)})+\mu t)),\end{array}\) 
and thus
\(W_{(k)} = \psi _{(k_1)}\phi _{(k)} k_1,\ k\in \Lambda _u\cup \Lambda _B,\ \ \ D_{(k)} = \psi _{(k_1)}\phi _{(k)} k_2,\ k\in \Lambda _B.\) 
Note that,
\(W_{(k)}\)  and \(D_{(k)}\)  are mean zero on 3.
Moreover, since
\({}_{\mathbb {T}^3} \phi _{(k)}^2(x) {\rm d}x = 1 ,\quad \text{and} \quad {}_{\mathbb {T}^3} \psi _{(k_1)}^2(x) {\rm d}x = 1,\) 
it follows that
\(\begin{array}{ll}&\displaystyle {}_{\mathbb {T}^3}W_{(k)}\otimes W_{(k)}{\rm d}x= k_1\otimes k_1,\ \ {}_{\mathbb {T}^3}D_{(k)}\otimes D_{(k)}{\rm d}x= k_2\otimes k_2,\end{array}\) 
\(\begin{array}{ll}&\displaystyle {}_{\mathbb {T}^3}W_{(k)}\otimes D_{(k)}{\rm d}x= k_1\otimes k_2,\ \ {}_{\mathbb {T}^3}D_{(k)}\otimes W_{(k)}{\rm d}x= k_2\otimes k_1.\end{array}\) 
It is worth noting that,
the building blocks in [2]}
have one oscillation direction \(k\) ,
orthogonal to the direction vectors \(k_1\)  and \(k_2\) ,
and have the \(1D\) -intermittency.
In order to control the strong viscosity and resistivity \((-\Delta )^{{\alpha }_i}\) 
when \({\alpha }_i\ge 1/2\) , \(i=1,2\) ,
the idea here to explore more intermittency,
inspired by the work [11]} in the NSE context,
is to introduce the new concentration function \(\psi _{(k_1)}\)  in (REF ).
Two new parameters \(r_{\perp }\)  (the rescaling parameter in the direction vertical to the flow)
and \(r_{\parallel }\)  (the rescaling parameter in the direction parallel to the flow)
parameterize the concentration of the intermittent flows \(W_{(k)}\)  and \(D_{(k)}\) .
In particular,
one has
\(\sum \limits _{k\in \Lambda _u\cup \Lambda _B} \Vert W_{(k)}\Vert _{C_tL^1_x}+ \sum \limits _{k\in \Lambda _B} \Vert D_{(k)}\Vert _{C_tL^1_x}\lesssim r_{\perp }^\frac{1}{2} r_{\parallel }^{\frac{1}{2}} = \lambda _{q+1}^{-1 + 4{\varepsilon }},\) 

which gives the almost \(2D\)  intermittency and thus
permits to control the fractional viscosity and resistivity \((-\Delta )^{{\alpha }_i}\)  in (REF )
for any \({\alpha }_1, {\alpha }_2\in [0,1)\) .
Furthermore,
the new parameter \(\mu \)  permits to control the temporal correctors later
to balance the high spatial oscillations,
arising from the concentration function \(\psi _{(k)}.\) 
More precisely,
using the orthogonality
\( \begin{array}{ll}&\nabla \phi _{(k)}\cdot k_1 = \nabla \phi _{(k)}\cdot k_2 =0, \\&\nabla \psi _{(k_1)}\cdot k = \nabla \psi _{(k_1)}\cdot k_2 =0,\end{array}\) 
we have the important algebraic identities
\(\begin{array}{ll}&\displaystyle {\mathrm {div}}( W_{(k)}\otimes W_{(k)}) =2(W_{(k)}\cdot \nabla \psi _{(k_1)})\phi _{(k)}k_1= \frac{1}{\mu }\partial _t \left(\psi _{(k_1)}^2 \phi _{(k)}^2 k_1\right), \\&\displaystyle {\mathrm {div}}( D_{(k)}\otimes D_{(k)}) =2(D_{(k)}\cdot \nabla \psi _{(k_1)})\phi _{(k)}k_2= 0,\end{array}\) 
and
\(\begin{array}{ll}& \displaystyle {\mathrm {div}}( D_{(k)}\otimes W_{(k)}) =({\mathrm {div}}W_{(k)}) D_{(k)}+(W_{(k)}\cdot \nabla )D_{(k)} = \frac{1}{\mu } \partial _t \left(\psi _{(k_1)}^2 \phi _{(k)}^2 k_2\right) , \\&\displaystyle {\mathrm {div}}( W_{(k)}\otimes D_{(k)}) =({\mathrm {div}}D_{(k)}) W_{(k)}+(D_{(k)}\cdot \nabla )W_{(k)} =0.\end{array}\) 
Thus, when combining with the temporal correctors,
module a pressure term,
the time derivative can be moved onto the low oscillating amplitudes
(see (REF ), (REF ) below).
Since the intermittent flow \(W_{(k)}\)  is not divergence-free,
we also need the corrector
\(\begin{aligned}\widetilde{W}_{(k)}^c := \frac{1}{\lambda ^2N_{ \Lambda }^2} \nabla \psi _{(k_1)}\times {\mathrm {curl}}(\Phi _{(k)} k_1) .\end{aligned}\) 
Then, straightforward computations show that
\(W_{(k)} + \widetilde{W}_{(k)}^c={\mathrm {curl}}{\mathrm {curl}}\left(\frac{1}{\lambda ^2N_{\Lambda }^2 } \psi _{(k_1)}\Phi _{(k)} k_1\right)={\mathrm {curl}}{\mathrm {curl}}W^c_{(k)},\) 
where \(W^c_{(k)}\)  is given by
\( W^c_{(k)} := \frac{1}{\lambda ^2N_{\Lambda }^2 } \psi _{(k_1)}\Phi _{(k)} k_1.\) 

Thus, it follows that
\( {\mathrm {div}}(W_{(k)}+ \widetilde{W}^c_{(k)}) =0.\) 
Regarding the magnetic flows,
we introduce the correctors
\(D_{(k)}^c:=\frac{1}{\lambda ^2N_{\Lambda }^2}\psi _{(k_1)}\Phi _{(k)}k_2,\quad \widetilde{D}_{(k)}^c:=-\frac{1}{ \lambda ^2N_\Lambda ^2}\Delta \psi _{(k_1)} \Phi _{(k)} k_2,\quad k\in \Lambda _B.\) 
It holds that
\( D_{(k)} + \widetilde{D}_{(k)}^c={\mathrm {curl}}{\mathrm {curl}}D^c_{(k)}.\) 
Lemma REF  below contains the key estimates of the intermittent
velocity and magnetic flows.
Lemma 3.3 (Estimates of spatial intermittency)  
For \(p \in [1,+\infty ]\) , \(N,\,M \in \mathbb {N}\) , we have
\(&\left\Vert \nabla ^{N} \partial _{t}^{M} \psi _{(k_1)}\right\Vert _{C_t L^{p}_{x}}\lesssim r_{\Vert }^{\frac{1}{p}- \frac{1}{2}}\left(\frac{r_{\perp } \lambda }{r_{\Vert }}\right)^{N}\left(\frac{r_{\perp } \lambda \mu }{r_{\Vert }}\right)^{M},  \\&\left\Vert \nabla ^{N} \phi _{(k)}\right\Vert _{L^{p}_{x}}+\left\Vert \nabla ^{N} \Phi _{(k)}\right\Vert _{L^{p}_{x}}\lesssim r_{\perp }^{\frac{1}{p}- \frac{1}{2}} \lambda ^{N}, \) 

where the implicit constants are independent of \(r_{\perp },\,r_{\parallel },\,\lambda \)  and \(\mu \) .
In particular, we have
\(&\displaystyle \left\Vert \nabla ^{N} \partial _{t}^{M} W_{(k)}\right\Vert _{C_t L^{p}_{x}}+\frac{r_{\Vert }}{r_{\perp }}\left\Vert \nabla ^{N} \partial _{t}^{M} \widetilde{W}_{(k)}^{c}\right\Vert _{C_t L^{p}_{x}}+\lambda ^{2}\left\Vert \nabla ^{N} \partial _{t}^{M} W_{(k)}^c\right\Vert _{C_t L^{p}_{x}}\displaystyle \nonumber \\&\qquad \lesssim r_{\perp }^{\frac{1}{p}- \frac{1}{2}} r_{\Vert }^{\frac{1}{p}- \frac{1}{2}} \lambda ^{N}\left(\frac{r_{\perp } \lambda \mu }{r_{\Vert }}\right)^{M}, \ \ k\in \Lambda _u \cup \Lambda _B,  \\&\displaystyle \left\Vert \nabla ^{N} \partial _{t}^{M} D_{(k)}\right\Vert _{C_t L^{p}_{x}}+\frac{r_{\Vert }}{r_{\perp }}\left\Vert \nabla ^{N} \partial _{t}^{M} \widetilde{D}_{(k)}^{c}\right\Vert _{C_t L^{p}_{x}}+\lambda ^{2}\left\Vert \nabla ^{N} \partial _{t}^{M} D_{(k)}^c\right\Vert _{C_t L^{p}_{x}} \nonumber \\&\qquad \lesssim r_{\perp }^{\frac{1}{p}- \frac{1}{2}} r_{\Vert }^{\frac{1}{p}- \frac{1}{2}}\lambda ^{N}\left(\frac{r_{\perp } \lambda \mu }{r_{\Vert }}\right)^{M},\ \ k\in \Lambda _B. \) 

By the definitions of \( \psi _{(k_1)}\) , \(\phi _{(k)}\)  and \(\Phi _{(k)}\) ,
the \(L^\infty \) -norms can be bounded by
\(&\left\Vert \nabla ^{N} \partial _{t}^{M} \psi _{(k_1)}\right\Vert _{C_{t,x}}\lesssim r_{\Vert }^{-\frac{1}{2}}\left(\frac{r_{\perp } \lambda }{r_{\Vert }}\right)^{N}\left(\frac{r_{\perp } \lambda \mu }{r_{\Vert }}\right)^{M}, \\&\left\Vert \nabla ^{N} \phi _{(k)}\right\Vert _{L^{\infty }_x}+\left\Vert \nabla ^{N} \Phi _{(k)}\right\Vert _{L^{\infty }_x}\lesssim \lambda ^{N}r_{\perp }^{-\frac{1}{2}}. \) 
In order to control the \(L^p\) -norm, \(1\le p<+{\infty }\) ,
let us estimate the volume of the support of \(\psi _{(k_1)}\)  below.
As \(\psi _{(k_1)}\)  is \((\mathbb {T}/(\lambda r_{\perp })^3)\) -periodic,
it suffices to estimate the volume of the support of \(\psi _{(k_1)}\)  on the cubes with side length \(2\pi /(\lambda r_{\perp })\) 
and then multiply the resulting estimate by \((\lambda r_{\perp })^3\) .
One may let the oscillation parameter \(\mu =0\)  and the shifts \({\alpha }_{(k)}=0\) , \(k\in \Lambda _u\cup \Lambda _B\) ,
without changing the volume of the support of \(\psi _{(k_1)}\) .
In each one of these cubes,
\(\psi _{(k_1)}\)  is supported on at most \(2N_{\Lambda }\)  thickened planes with thickness
\(\sim {r_{\parallel }}/{(\lambda r_{\perp })}\) ,
which are parallel to each other and perpendicular to the direction \(k_1\) .
Hence,
the volume of the support of \(\psi _{(k_1)}\)  on \(\mathbb {T}^3\)  can be bounded by
\(|\mathrm {supp\,}\psi _{(k_1)}|\lesssim (\lambda r_{\perp })^{-2}\frac{r_{\parallel }}{\lambda r_{\perp }} (\lambda r_{\perp })^3\lesssim r_{\parallel }.\) 
This along with (REF ) yields that
\(\Vert \nabla ^{N} \partial _t^M \psi _{(k_1)}\Vert _{C_t L^p_x}\lesssim |\mathrm {supp\,}\psi _{(k_1)}|^{\frac{1}{p}} \left\Vert \nabla ^{N} \partial _t^M \psi _{(k_1)}\right\Vert _{C_t L^{\infty }_x}\lesssim r_{\parallel }^{\frac{1}{p}-\frac{1}{2}} \left(\frac{r_{\perp }\lambda }{r_{\parallel }}\right)^N \left(\frac{r_{\perp }\lambda \mu }{r_{\parallel }}\right)^M.\) 

Since there are only finite wavevectors in \(\Lambda _u\)  and \(\Lambda _B\) ,
the implicit constant can be taken independent of \(k_1\)  by taking the maximum over the wavevector sets
\(\Lambda _u\)  and \(\Lambda _B\) .
Thus, (REF ) is verified.
Estimate () can be proved similarly.
Actually,
in each of the cubic with length \(2\pi /(\lambda r_{\perp })\) ,
\(\phi _{(k)}\)  is supported on at most \(2N_{\Lambda }\)  thickened planes
with thickness \(\sim \lambda ^{-1}\) ,
perpendicular to the direction \(k\) .
Hence,
\(|\mathrm {supp\,}\phi _{(k)}| \lesssim (\lambda r_{\perp })^{-2} \lambda ^{-1} (\lambda r_{\perp })^3\lesssim r_{\perp },\) 

which along with () yields that
\(\Vert {\nabla }^N \phi _{(k)}\Vert _{L^p_x}\lesssim |\mathrm {supp\,}\phi _{(k)}|^{\frac{1}{p}} \Vert {\nabla }^N \phi _{(k)}\Vert _{L^{\infty }_x}\lesssim r_{\perp }^{\frac{1}{p}} \lambda ^N r_{\perp }^{-\frac{1}{2}}\lesssim r_{\perp }^{\frac{1}{p}-\frac{1}{2}} \lambda ^N.\) 

Thus, () is verified.
Finally, estimates (REF ) and () follow from (REF ), ()
and Fubini's theorem.

Temporal building blocks.
As mentioned in Section 1, the previous construction of intermittent velocity and magnetic flows
cannot provide us with enough intermittency to handle the
stronger viscosity and resistivity \((-\Delta )^{{\alpha }_i}\)  when \({\alpha }_i \in [1,5/4)\) , \(i=1,2\) .
The key idea is to exploit more intermittency
from temporal oscillations.
Two new parameters \(\tau \)  and \(\sigma \)  will be introduced to
parameterize the concentrations of
the temporal function.
More precisely, let \(g\in C_c^\infty (\)  be any cut-off function such that
\({}_{ g^2(t) {\rm d}t=1.}Let \tau \in \mathbb {N}_+ and rescale the cut-off function g by{\begin{@align}{1}{-1}g_\tau (t)=\tau ^{\frac{1}{2}} g(\tau t).\end{@align}}Then, we periodize g_\tau  such that the resulting functions(by an abuse of notion, still denoted by g_\tau )are periodic functions.\) 
In order to construct the temporal correctors \(w_{q+1}^{(o)}\)  and \(d_{q+1}^{(o)}\)  (see (REF )-())
to balance the high temporal oscillations
arising from the concentration function \(g_{(\tau )}\) ,
we also need the function
\(h_\tau : \mathbb {T} \rightarrow {\mathbb {R}}\) , defined by
\( h_\tau (t):= \int _{-\pi }^t \left(g_\tau ^2(s) - 1\right)\ ds.\) 
Set
\(g_{(\tau )}:=g_\tau (\sigma t),\ \ h_{(\tau )}(t):= h_\tau (\sigma t).\) 
Note that, \(h_{(\tau )}\)  has the uniform \(L_t^{\infty }\) -bound
\(\Vert h_{(\tau )}\Vert _{L_t^\infty (}\le 1.\) 

Moreover,
we have the following temporal intermittent estimates of \(g_{(\tau )}\) .
Lemma 3.4 (Estimates of temporal intermittency)  
For \(\gamma \in [1,+\infty ]\) , \(M \in \mathbb {N}\) ,
we have
\(\left\Vert \partial _{t}^{M}g_{(\tau )}\right\Vert _{L^{\gamma }_t} \lesssim \sigma ^{M}\tau ^{M+\frac{1}{2}-\frac{1}{\gamma }},\) 

where the implicit constants are independent of \(\tau \)  and \(\sigma \) .

As in the proof of Lemma REF ,
we first estimate the \(L^\infty \) -norm of \(g_{(\tau )}\) .
By definition,
\(\left\Vert \partial _{t}^{M}g_{(\tau )}\right\Vert _{L^{{\infty }}_t(} \lesssim \sigma ^{M}\tau ^{M+\frac{1}{2}}.\) 

Next, we give a bound on the volume of the support of \(g_{(\tau )}\)  on \(.Since \) g()\( is \)\(-periodic,it is sufficient to obtain the estimate on the interval \) [-/,/]\(and then multiply the result by \)\(.Note that, in the interval \) [-/,/]\(,\) g()\( is supported on an interval with length \) ()-1\(,and thus the volume of the support of \) g()\( on \) 
is bounded by
\(|\mathrm {supp\,}g_{(\tau )}| \lesssim (\sigma \tau )^{-1} \sigma \lesssim \tau ^{-1}.\) 

Combining (REF ) and (REF ) together we obtain
\(\Vert  \partial _t^M g_{(\tau )}\Vert _{L^\gamma _t} \lesssim |\mathrm {supp\,}g_{(\tau )}|^{\frac{1}{\gamma }} \left\Vert  \partial _t^M g_{(\tau )}\right\Vert _{L^{\infty }_t(}\lesssim \sigma ^{M}\tau ^{M+\frac{1}{2}-\frac{1}{\gamma }}.\) 

Therefore, estimate (REF ) is proved.

Velocity and magnetic perturbations
This section is devoted to the construction of the key velocity and magnetic perturbations,
including the principal parts,
the incompressibility correctors,
the spatial and temporal correctors.
To begin with,
let us first define suitable amplitudes of perturbations,
which are the key to decrease the effects of old velocity and magnetic stresses.
Amplitudes
The magnetic amplitudes.
Let \(\chi : [0, +\infty ) \rightarrow \mathbb {R}\)  be a smooth cut-off function such that
\(\chi (z) =\left\lbrace \aligned & 1,\quad 0 \le z\le 1, \\& z,\quad z \ge 2,\endaligned \right.\) 
and
\(\frac{1}{2} z \le \chi (z) \le 2z \quad \text{for}\quad z \in (1,2).\) 
Set
\(\rho _B(t,x) := 2 \varepsilon _B^{-1} \delta _{q+ 1} \chi \left( \frac{|\mathring{R}_{\ell }^B(t, x) |}{\delta _{q+1} } \right),\) 
where \(\varepsilon _{B}\)  is the small radius in the geometric Lemma REF .
Then, by (REF ), (REF ) and (REF ),
\(\left| \frac{\mathring{R}_{\ell }^B}{\rho _B} \right|= \left| \frac{\mathring{R}_{\ell }^B}{2 \varepsilon _B^{-1} \delta _{q+ 1}\chi ( \delta _{q+1} ^{-1} |\mathring{R}_{\ell }^B | )} \right| \le \varepsilon _B,\) 
and for any \(p\in [1,+\infty ]\) ,
\(&\rho _B\ge \varepsilon _B^{-1} \delta _{q+ 1},\\&\Vert  \rho _B \Vert _{L^p_{t,x}} \le 8 \varepsilon _B^{-1} \left( (16\pi ^4)^{\frac{1}{p}}\delta _{q+1} + \Vert \mathring{R}_{\ell }^{B} \Vert _{L^p_{t,x}} \right).\) 

Furthermore, by the standard Hölder estimate (see [36]}),
the iterative estimate () and (REF ), for \(1\le N\le 4\) ,
\(& \Vert  \rho _B \Vert _{C_{t,x}} \lesssim \ell ^{-1} , \quad \Vert  \rho _B \Vert _{C_{t,x}^N} \lesssim \ell ^{-N}\delta ^{-N+1}_{q+1}, \\&\Vert  \rho _B^{1/2}\Vert _{C_{t,x}} \lesssim \ell ^{-1}, \quad \Vert  \rho _B^{1/2} \Vert _{C_{t,x}^N} \lesssim \ell ^{-N} \delta ^{-2N}_{q+1},  \\&\Vert  \rho _B^{-1}\Vert _{C_{t,x}}\lesssim \delta _{q+ 1}^{-1} ,\quad \Vert  \rho _B^{-1} \Vert _{C_{t,x}^N } \lesssim \ell ^{-N}\delta ^{-2N}_{q+1} . \) 
Moreover,
we choose the smooth temporal cut-off function \(f_B\)  such that

\(0\le f_B\le 1\) , \(f_B \equiv 1\)  on \(\mathrm {supp\,}_t \mathring{R}_{\ell }^B\) ;

\(\mathrm {supp\,}_t f_B \subseteq N_l(\mathrm {supp\,}_t \mathring{R}_{\ell }^B)\) ;

\(\Vert f_B\Vert _{C_t^N}\lesssim \ell ^{-N}\) ,  \(1\le N\le 4\) .

The amplitudes of the magnetic perturbations are defined by
\(a_{(k)}(t,x):= a_{k, B}(t,x)= \rho _B^{\frac{1}{2} } (t,x) f_B(t)\gamma _{(k)}\left(\frac{-\mathring{R}_{\ell }^B(t,x)}{\rho _B(t,x)}\right), \quad k \in \Lambda _B,\) 
where \(\gamma _{(k)}\)  is the smooth function in the geometric Lemma REF .
Note that, by virtue of the geometric Lemma REF ,
the identity (REF )
and the expression (REF ) of \(a_{(k)}\) ,
the algebraic identity holds:
\(& \sum \limits _{ k \in \Lambda _B} a_{(k)}^2 g_{(\tau )}^2( D_{(k)} \otimes W_{(k)} - W_{(k)} \otimes D_{(k)} ) \\= & -\mathring{R}_{\ell }^B+ \sum \limits _{ k \in \Lambda _B} a_{(k)}^2g_{(\tau )}^2\mathbb {P}_{\ne 0}( D_{(k)} \otimes W_{(k)} -W_{(k)} \otimes D_{(k)} ) \\& + \sum _{k \in \Lambda _B} a_{(k)}^2 (g_{(\tau )}^2-1) {}_{3}D_{(k)}\otimes W_{(k)}-W_{(k)}\otimes D_{(k)}{\rm d}x ,\) 

where \(\mathbb {P}_{\ne 0}\)  denotes the spatial projection onto nonzero Fourier modes.
Moreover, in view of (REF )-(), we also have the following analytic estimates.
Lemma 4.1 (Estimates of magnetic amplitudes)  
For \(1\le N\le 4\) , \(k\in \Lambda _B\) , we have
\(&\Vert a_{(k)}\Vert _{L^2_{t,x}} \lesssim \delta _{q+1}^{\frac{1}{2}} ,\\& \Vert  a_{(k)} \Vert _{C_{t,x}} \lesssim \ell ^{-1},\ \ \Vert  a_{(k)} \Vert _{C_{t,x}^N} \lesssim \ell ^{-4N}.\) 

To obtain the \(L^2_{t,x}\) -estimate of \(a_{(k)}\) , \(k\in \Lambda _{B}\) ,
we use () and () to bound
\(\Vert a_{(k)}\Vert _{L^2_{t,x}}&\le \Vert  \rho _B \Vert _{L^1_{t,x}}^{\frac{1}{2}} \Vert  \gamma _{(k)} \Vert _{C_x(B_{\varepsilon _B}(0))} \Vert f_B\Vert _{C_t} \\&\le M_* (8 \varepsilon _B^{-1})^{\frac{1}{2}}(16\pi ^4 \delta _{q+1} + \Vert \mathring{R}_{\ell }^{B} \Vert _{L^1_{t,x}})^{\frac{1}{2}} \\&\le M_* (8 \varepsilon _B^{-1})^{\frac{1}{2}}( 16\pi ^4 + 1)^{\frac{1}{2}} \delta _{q+1}^{\frac{1}{2}},\) 

which yields (REF ).
Moreover, for \(N\ge 1\) ,
by () and the standard Hölder estimate (cf. [36]}),
\(\Vert a_{(k)}\Vert _{C^N_{t,x}}&\lesssim \sum _{0\le N_1+N_2+N_3\le N}\Vert \rho _B^{\frac{1}{2}} \Vert _{C^{N_1}_{t,x}}\Vert  \gamma _{(k)} (-\rho _B^{-1} \mathring{R}^B_{\ell } ) \Vert _{C^{N_2}_{t,x}} \Vert f_B\Vert _{C_t^{N_3}} \\&\lesssim \sum _{0\le N_1+N_2+N_3\le N} \ell ^{-N_1-1}\ell ^{-N_3}\left(1+ \Vert \rho _B^{-1}\mathring{R}^B_{\ell }\Vert _{C^{N_2}_{t,x}} + \Vert \rho _B^{-1}\mathring{R}^B_{\ell }\Vert _{C_{t,x}}^{N_2-1}\Vert \rho _B^{-1}\mathring{R}^B_{\ell }\Vert _{C^{N_2}_{t,x}}\right) .\) 

Since by (), () and (),
\(\Vert \rho _B^{-1}\mathring{R}^B_{\ell }\Vert _{C^{N_2}_{t,x}}&\lesssim \sum _{0\le N_{21}+N_{22} \le N_2}\Vert \rho _B^{-1}\Vert _{C^{N_{21}}_{t,x}}\Vert \mathring{R}^B_{\ell }\Vert _{C^{N_{22}}_{t,x}} \nonumber \\& \lesssim \sum _{0\le N_{21}+ N_{22}\le N_2}\ell ^{-N_{21}-1}\ell ^{-N_{22}}\lesssim \ell ^{-N_{2}-1},\) 

we get
\(\Vert a_{(k)}\Vert _{C^N_{t,x}} &\lesssim \sum _{0\le N_1+N_2+N_3\le N}\ell ^{-N_1-1}\ell ^{-N_3}(\ell ^{-N_2-1}+\ell ^{-2N_2} )\lesssim \ell ^{-4N}.\) 
The \(C_{t,x}\) -estimate of \(a_{(k)}\)  can be bounded by,
via (),
\(\Vert a_{(k)}\Vert _{C_{t,x}}&\lesssim \Vert  \rho _B^{\frac{1}{2}} \Vert _{C_{t,x}}\Vert f_B\Vert _{C_t} \Vert  \gamma _{(k)} \Vert _{C_x(B_{\varepsilon _B}(0))}\lesssim \ell ^{-1}.\) 

Therefore, the estimates in () are verified.
The proof is complete.

The velocity amplitudes.
Below we define the velocity amplitudes.
Unlike in the previous magnetic case,
because of the strong coupling between the velocity and magnetic fields,
a new matrix \(\mathring{G}^{B}\)  is needed
in order to maintain the cancellation between the perturbations and the old stresses,
\(\mathring{G}^{B}: = \sum _{k \in \Lambda _B}a_{(k)}^2 {}_{\mathbb {T}^3} W_{(k)} \otimes W_{(k)} - D_{(k)} \otimes D_{(k)}{\rm d}x.\) 
In view of estimates (REF ) and (),
we have that for \(N\ge 1\) ,
\(\Vert \mathring{G}^{B}\Vert _{C_{t,x}} \lesssim \ell ^{-2},\quad \Vert \mathring{G}^{B}\Vert _{C_{t,x}^N} \lesssim \ell ^{-4N-1} ,\quad \Vert \mathring{G}^{B}\Vert _{L^1_{t,x}} &\lesssim \delta _{q+1}.\) 
Set
\(\begin{aligned}& \rho _u(t,x):= 2 \varepsilon _u^{-1} \delta _{q+ 1}\chi \left( \frac{|\mathring{R}_{\ell }^u(t,x) + \mathring{G}^{B}(t,x)|}{\delta _{q+1}} \right).\end{aligned}\) 
Then, by (REF ) and (REF ),
\( & \left| \frac{\mathring{R}_{\ell }^u + \mathring{G}^{B}}{\rho _u} \right| \le \varepsilon _u.\) 
We choose the smooth temporal cut-off function \(f_u\)  such that

\(0\le f_u\le 1\) , \(f_u\equiv 1\)  on \(\mathrm {supp\,}_t \mathring{R}_{\ell }^u\cup \mathrm {supp\,}_t \mathring{G}^B\) ;

\(\mathrm {supp\,}_t f_u\subseteq N_l(\mathrm {supp\,}_t \mathring{R}_{\ell }^u\cup \mathrm {supp\,}_t \mathring{G}^B)\subseteq N_{2l}(\mathrm {supp\,}_t \mathring{R}_{\ell }^u\cup \mathrm {supp\,}_t \mathring{R}_{\ell }^B)\) ;

\(\Vert f_u\Vert _{C_t^N}\lesssim \ell ^{-N}\) , \(1\le N\le 4\) .

The velocity amplitudes are defined by
\(\begin{aligned}&a_{(k)} (t,x) := a_{k, u}(t, x)= \rho _u^{\frac{1}{2}} f_u(t)\gamma _{(k)}\left(\mathrm {Id}- \frac{\mathring{R}_{\ell }^u(t,x) + \mathring{G}^{B}(t,x)}{\rho _u(t,x)} \right),\quad k \in \Lambda _u.\end{aligned}\) 
In view of the normalization (REF ), the geometric Lemma REF 
and the expression (REF ) of \(a_{(k)}\) , we get the following algebraic identity:
\(\sum \limits _{ k \in \Lambda _u} a_{(k)}^2 g_{(\tau )}^2W_{(k)} \otimes W_{(k)}& = \rho _u f^2_u \mathrm {Id}- \mathring{R}_{\ell }^u - \mathring{G}^{B}+ \sum \limits _{ k \in \Lambda _u} a_{(k)}^2 g_{(\tau )}^2 \mathbb {P}_{\ne 0}( W_{(k)} \otimes W_{(k)} )\\& \quad +\sum _{k\in \Lambda _u}a_{(k)}^{2}\left(g_{(\tau )}^2-1 \right){}_{3}W_{(k)}\otimes W_{(k)}{\rm d}x .\) 
Moreover, similarly to (REF )-(),
we have for any \(p \in [1, +\infty )\) ,
\(&\rho _u\ge \varepsilon _u^{-1} \delta _{q+ 1}, \\&\Vert  \rho _u \Vert _{L^p_{t,x}} \le 8 \varepsilon _u^{-1}\left( (16\pi ^4)^{\frac{1}{p}} \delta _{q+1} + \Vert \mathring{R}_{\ell }^u + \mathring{G}^{B}\Vert _{L^p_{t,x}} \right),\) 

and for \(1\le N\le 4\) ,
\(& \Vert  \rho _u \Vert _{C_{t,x}} \lesssim \ell ^{-2} , \quad \Vert  \rho _u \Vert _{C_{t,x}^N} \lesssim \ell ^{-5N} \delta _q^{-N+1},  \\&\Vert  \rho _u^{1/2}\Vert _{C_{t,x} } \lesssim \ell ^{-1}, \quad \Vert  \rho _u^{1/2} \Vert _{C_{t,x}^N} \lesssim \ell ^{-5N} \delta _{q+1}^{-2N} ,  \\&\Vert  \rho _u^{-1}\Vert _{C_{t,x}}\lesssim \delta _{q+ 1}^{-1} , \quad \Vert  \rho _u^{-1} \Vert _{C_{t,x}^N } \lesssim \ell ^{-5N}\delta _{q+1}^{-2N} . \) 
Arguing as in the proof of Lemma REF ,
using the standard Hölder estimate (cf. [36]})
and estimates (), (REF ), (REF )-()
we have the following analytic estimates for the velocity amplitude functions.
Lemma 4.2 (Estimates of velocity amplitudes)  
For \(1\le N\le 4\) , \(k\in \Lambda _{u} \) ,
we have
\(&\Vert a_{(k)}\Vert _{L^2_{t,x}} \lesssim \delta _{q+1}^{\frac{1}{2}} ,\\& \Vert  a_{(k)} \Vert _{C_{t,x}} \lesssim \ell ^{-1},\quad \Vert a_{(k)} \Vert _{C_{t,x}^N} \lesssim \ell ^{-8N}.\) 


Principal parts of perturbations
We define the principal parts \(w_{q+1}^{(p)}\)  and \(d_{q+1}^{(p)}\) ,
respectively,
of the velocity
and magnetic perturbations by
\(w_{q+1}^{(p)} &:= \sum _{k \in \Lambda _u \cup \Lambda _B } a_{(k)}g_{(\tau )}W_{(k)},\\d_{q+1}^{(p)} &:= \sum _{k \in \Lambda _B} a_{(k)}g_{(\tau )}D_{(k)} .\) 
By the orthogonality \(W_{(k)} \otimes D_{(k^{\prime })} =0\) , \(k\ne  k^{\prime }\) ,
and the algebraic identity (REF ),
\( & d_{q+ 1}^{(p)} \otimes w_{q+ 1}^{(p)} - w_{q+1}^{(p)}\otimes d_{q+ 1}^{(p)}+ \mathring{R}^B_{\ell } \nonumber \\=& \sum _{k \in \Lambda _B} a_{(k)}^2g_{(\tau )}^2 (D_{(k)}\otimes W_{(k)}-W_{(k)}\otimes D_{(k)}) + \mathring{R}^B_{\ell } \nonumber \\=& \sum _{k \in \Lambda _B} a_{(k)}^2 g_{(\tau )}^2 \mathbb {P}_{\ne 0}(D_{(k)}\otimes W_{(k)}-W_{(k)}\otimes D_{(k)}) \nonumber \\&+ \sum _{k \in \Lambda _B} a_{(k)}^2 (g_{(\tau )}^2-1) {}_{3}D_{(k)}\otimes W_{(k)}-W_{(k)}\otimes D_{(k)}{\rm d}x .\) 
Moreover, for the nonlinearity in the velocity equation,
by (REF ) and (REF ),
\( & w_{q+ 1}^{(p)} \otimes w_{q+ 1}^{(p)} - d_{q+ 1}^{(p)} \otimes d_{q+1}^{(p)} + \mathring{R}^u_\ell \nonumber \\=& \sum _{k \in \Lambda _u} a_{(k)}^2g_{(\tau )}^2 W_{(k)}\otimes W_{(k)}+ \sum _{k \in \Lambda _B} a_{(k)}^2 g_{(\tau )}^2(W_{(k)}\otimes W_{(k)}-D_{(k)}\otimes D_{(k)}) + \mathring{R}^u_\ell \nonumber \\=& \rho _{u} f^2_u Id +\sum _{k \in \Lambda _u} a_{(k)}^2 g_{(\tau )}^2\mathbb {P}_{\ne 0}( W_{(k)}\otimes W_{(k)}) \nonumber \\& + \sum _{k \in \Lambda _B} a_{(k)}^2 g_{(\tau )}^2\mathbb {P}_{\ne 0} (W_{(k)}\otimes W_{(k)}-D_{(k)}\otimes D_{(k)})\nonumber \\& +\sum _{k\in \Lambda _u}a_{(k)}^{2}\left(g_{(\tau )}^2-1 \right){}_{3}W_{(k)}\otimes W_{(k)}{\rm d}x \nonumber \\&+ \sum _{k\in \Lambda _B}a_{(k)}^{2}\left(g_{(\tau )}^2-1 \right){}_{3}W_{(k)}\otimes W_{(k)}-D_{(k)}\otimes D_{(k)}{\rm d}x.\) 
Remark 4.3  The key fact here is that,
module the high frequency part,
the nonlinearities of the principal parts
cancel with the old magnetic stresses \(\mathring{R}_{\ell }^B\)  and \(\mathring{R}_{\ell }^u\) ,
which enables to decrease the amplitudes of the old stresses
and to yield the inductive estimate ().


Incompressibility correctors
Because the amplitude functions \(\lbrace a_{(k)}, k\in \Lambda _u \cup \Lambda _B\rbrace \)  depend on the space,
the principal parts of perturbations are not divergence free.
This leads to the introduction of the incompressibility correctors
\(w_{q+1}^{(c)}&:= \sum _{k\in \Lambda _u \cup \Lambda _B } g_{(\tau )}\left({\mathrm {curl}}(\nabla a_{(k)} \times W^c_{(k)})+ \nabla a_{(k)} \times {\mathrm {curl}}W^c_{(k)} +a_{(k)} \widetilde{W}_{(k)}^c \right) ,  \\d_{q+1}^{(c)} &:= \sum _{k\in \Lambda _B } g_{(\tau )}\left({\mathrm {curl}}(\nabla a_{(k)} \times D_{(k)}^c)+ \nabla a_{(k)} \times {\mathrm {curl}}D_{(k)}^c+a_{(k)}\widetilde{D}_{(k)}^c\right), \) 


where \(W^c_{(k)}\)  and \(\widetilde{W}_k^c \)  are given by (REF ) and (REF ), respectively,
and
\(D_{(k)}^c\)  and \(\widetilde{D}_{(k)}^c\)  are as in (REF ).
Then, it follows from (REF ) and (REF ) that
\(& w_{q+1}^{(p)} + w_{q+1}^{(c)}={\mathrm {curl}}{\mathrm {curl}}\left( \sum _{k \in \Lambda _u \cup \Lambda _B} a_{(k)} g_{(\tau )}W^c_{(k)} \right),  \\& d_{q+1}^{(p)} + d_{q+1}^{(c)}={\mathrm {curl}}{\mathrm {curl}}\left( \sum _{k \in \Lambda _B} a_{(k)}g_{(\tau )}D_{(k)}^c \right). \) 


In particular,
\( {\mathrm {div}}(w_{q+1}^{(p)} + w_{q +1}^{(c)}) = {\mathrm {div}}(d_{q+1}^{(p)} + d_{q +1}^{(c)}) = 0,\) 

which justifies the definition of the incompressibility correctors.

Temporal correctors
Two new types of temporal correctors will be introduced
in order to balance
the high spatial and temporal oscillations in (REF ) and (REF ).
Temporal correctors to balance spatial oscillations.
In order to balance the high spatial oscillations in (REF ) and (REF )
caused by the spatial concentration function \(\psi _{(k_1)}\) ,
we introduce the temporal correctors \(w_{q+1}^{(t)}\)  and \(d_{q+1}^{(t)}\) ,
defined by
\(&w_{q+1}^{(t)} := -{\mu }^{-1} \sum _{k\in \Lambda _u\cup \Lambda _B} \mathbb {P}_{H}\mathbb {P}_{\ne 0}(a_{(k)}^2g_{(\tau )}^2 \psi _{(k_1)}^2 \phi _{(k)}^2 k_1),\\&d_{q+1}^{(t)}= -{\mu }^{-1} \sum _{k\in \Lambda _B}\mathbb {P}_{H}\mathbb {P}_{\ne 0}(a_{(k)}^2g_{(\tau )}^2 \psi _{(k_1)}^2 \phi _{(k)}^2 k_2),\) 


where \(\mathbb {P}_{H}\)  denotes the Helmholtz-Leray projector, i.e.,
\(\mathbb {P}_{H}=\mathrm {Id}-\nabla \Delta ^{-1}{\mathrm {div}}. \)
Then, by the Leibniz rule,
(REF ) and (REF ),
\( &\partial _{t} w_{q+1}^{(t)}+ \sum _{k \in \Lambda _u \cup \Lambda _B} \mathbb {P}_{\ne 0}\left(a_{(k)}^{2}g_{(\tau )}^2 {\mathrm {div}}(W_{(k)} \otimes W_{(k)})\right) \nonumber \\=&- {\mu }^{-1} \sum _{k \in \Lambda _u \cup \Lambda _B} \mathbb {P}_{H} \mathbb {P}_{\ne 0} \partial _{t}\left(a_{(k)}^{2}g_{(\tau )}^2 \psi _{(k_1)}^{2} \phi _{(k)}^{2} k_1\right) \nonumber \\& + {\mu }^{-1} \sum _{k \in \Lambda _u \cup \Lambda _B} \mathbb {P}_{\ne 0}\left(a_{(k)}^{2}g_{(\tau )}^2 \partial _{t}(\psi _{(k_1)}^{2} \phi _{(k)}^{2} k_1)\right) \nonumber \\=&(\nabla \Delta ^{-1}{\mathrm {div}}) {\mu }^{-1} \sum _{k \in \Lambda _u \cup \Lambda _B} \mathbb {P}_{\ne 0} \partial _{t}\left(a_{(k)}^{2}g_{(\tau )}^2 \psi _{(k_1)}^{2} \phi _{(k)}^{2} k_1\right) \nonumber \\& - {\mu }^{-1} \sum _{k \in \Lambda _u \cup \Lambda _B} \mathbb {P}_{\ne 0}(\partial _{t}\left( a_{(k)}^{2}g_{(\tau )}^2) \psi _{(k_1)}^{2} \phi _{(k)}^{2} k_1\right),\) 

and
\( &\partial _{t} d_{q+1}^{(t)}+ \sum _{k\in \Lambda _B} \mathbb {P}_{\ne 0}\left(a_{(k)}^{2}g_{(\tau )}^2 {\mathrm {div}}(D_{(k)} \otimes W_{(k)}-W_{(k)} \otimes D_{(k)} )\right) \nonumber \\=&(\nabla \Delta ^{-1}{\mathrm {div}}) {\mu }^{-1} \sum _{k \in \Lambda _B} \mathbb {P}_{\ne 0} \partial _{t}\left(a_{(k)}^{2}g_{(\tau )}^2 \psi _{(k_1)}^{2} \phi _{(k)}^{2} k_2\right) \nonumber \\& - {\mu }^{-1} \sum _{k \in \Lambda _B}\mathbb {P}_{\ne 0}(\partial _{t} \left(a_{(k)}^{2}g_{(\tau )}^2) \psi _{(k_1)}^{2}\phi _{(k)}^{2} k_2 \right).\) 
Remark 4.4  The important fact here is that,
the first terms on the R.H.S. of (REF ) and (REF )
can be treated as the pressures
and can be removed by using the Helmholtz-Leray projector,
while the remaining terms are of low spatial oscillations
and thus can be controlled by the large parameter \(\mu \) .
See (REF ) and (REF ) below.


Temporal correctors to balance temporal oscillations.
Another new type of temporal correctors
is introduced below,
in order to balance the high temporal oscillations
in (REF ) and (REF )
caused by the temporal concentration function \(g_{(\tau )}\) ,
\(& w_{q+1}^{(o)}:= -\sigma ^{-1}\sum _{k\in \Lambda _u }\mathbb {P}_{H}\mathbb {P}_{\ne 0}\left(h_{(\tau )}{}_{3} W_{(k)}\otimes W_{(k)}{\rm d}x\nabla (a_{(k)}^2) \right)\\&\qquad \quad \ -\sigma ^{-1}\sum _{k\in \Lambda _B }\mathbb {P}_{H}\mathbb {P}_{\ne 0}\left(h_{(\tau )}{}_{3} W_{(k)}\otimes W_{(k)}-D_{(k)}\otimes D_{(k)}{\rm d}x\nabla (a_{(k)}^2)\right) ,\\& d_{q+1}^{(o)}:= -\sigma ^{-1}\sum _{k\in \Lambda _B }\mathbb {P}_{H}\mathbb {P}_{\ne 0}\left(h_{(\tau )} ){}_{3} D_{(k_)}\otimes W_{(k)}-W_{(k)}\otimes D_{(k)}{\rm d}x\nabla (a_{(k)}^2)\right), \) 

where we recall that \(h_{(\tau )}\)  is given by (REF ).
Then, by the Leibniz rule
and the fact that \(\frac{d}{dt}h_{(\tau )}= \sigma (g^2_{(\tau )}-1)\) ,
\( &\partial _{t} w_{q+1}^{(o)}+\sum _{k\in \Lambda _u}\mathbb {P}_{\ne 0}\left(\left(g_{(\tau )}^2-1 \right){}_{3}W_{(k)}\otimes W_{(k)}{\rm d}x \nabla (a_{(k)}^{2}) \right) \nonumber \\&+ \sum _{k\in \Lambda _B}\mathbb {P}_{\ne 0}\left( \left(g_{(\tau )}^2-1 \right){}_{3}W_{(k)}\otimes W_{(k)}-D_{(k)}\otimes D_{(k)}{\rm d}x\nabla (a_{(k)}^2)\right) \nonumber \\=&\left(\nabla \Delta ^{-1}{\mathrm {div}}\right) \sigma ^{-1} \sum _{k \in \Lambda _u} \mathbb {P}_{\ne 0} \partial _{t}\left(h_{(\tau )}{}_{3}W_{(k)}\otimes W_{(k)}{\rm d}x\nabla (a_{(k)}^2)\right) \nonumber \\&+\left(\nabla \Delta ^{-1}{\mathrm {div}}\right) \sigma ^{-1} \sum _{k \in \Lambda _B} \mathbb {P}_{\ne 0} \partial _{t}\left(h_{(\tau )}{}_{3}W_{(k)}\otimes W_{(k)}-D_{(k)}\otimes D_{(k)}{\rm d}x\nabla (a_{(k)}^2)\right) \nonumber \\&-\sigma ^{-1}\sum _{k\in \Lambda _u}\mathbb {P}_{\ne 0}\left(h_{(\tau )}{}_{3}W_{(k)}\otimes W_{(k)}{\rm d}x\partial _t\nabla (a_{(k)}^2)\right) \nonumber \\&-\sigma ^{-1}\sum _{k\in \Lambda _B}\mathbb {P}_{\ne 0}\left(h_{(\tau )}{}_{3}W_{(k)}\otimes W_{(k)}-D_{(k)}\otimes D_{(k)}{\rm d}x\partial _t\nabla (a_{(k)}^2)\right),\) 

and
\( &\partial _{t} d_{q+1}^{(o)}+\sum _{k\in \Lambda _B}\mathbb {P}_{\ne 0}\left(\left(g_{(\tau )}^2-1 \right){}_{3}D_{(k)}\otimes W_{(k)}-W_{(k)}\otimes D_{(k)}{\rm d}x\nabla (a_{(k)}^{2})\right) \nonumber \\=&\left(\nabla \Delta ^{-1}{\mathrm {div}}\right) \sigma ^{-1} \sum _{k \in \Lambda _B} \mathbb {P}_{\ne 0} \partial _{t}\left(h_{(\tau )}{}_{3}D_{(k)}\otimes W_{(k)}-W_{(k)}\otimes D_{(k)}{\rm d}x\nabla (a_{(k)}^{2})\right) \nonumber \\&-\sigma ^{-1}\sum _{k\in \Lambda _B}\mathbb {P}_{\ne 0}\left(h_{(\tau )}{}_{3}D_{(k)}\otimes W_{(k)}-W_{(k)}\otimes D_{(k)}{\rm d}x\partial _t\nabla (a_{(k)}^{2})\right) .\) 
Remark 4.5  As in the previous spatial oscillation case in Remark REF ,
the first two terms on the R.H.S. of (REF )
are the harmless pressures,
and the remaining terms are of low temporal oscillations
which can be controlled by the large parameter \(\sigma \) .
Similar structure also appear in the formula (REF ).


Velocity and magnetic perturbations
We are now in position to define the velocity and magnetic perturbations \(w_{q+1}\)  and \(d_{q+1}\) 
at level \(q+1\) :
\(w_{q+1} &:= w_{q+1}^{(p)} + w_{q+1}^{(c)}+ w_{q+1}^{(t)}+w_{q+1}^{(o)}, \\d_{q+1} &:= d_{q+1}^{(p)} + d_{q+1}^{(c)}+ d_{q+1}^{(t)}+d_{q+1}^{(o)}.\) 


Then, the velocity and magnetic fields at level \(q+1\)  are defined by
\(& u_{q+1}:= u_{\ell } + w_{q+1},\\& B_{q+1}:= B_{\ell }+ d_{q+1} .\) 
The estimates of velocity and magnetic perturbations are summarized in Lemma REF .
Lemma 4.6 (Estimates of perturbations)  
For any \(\rho \in (1,{\infty }), \gamma \in [1,+\infty ]\)  and integer \(0\le N\le 4\) ,
the following estimates hold:
\(&\Vert {\nabla }^N w_{q+1}^{(p)} \Vert _{L^ \gamma _tL^\rho _x } + \Vert {\nabla }^N d_{q+1}^{(p)} \Vert _{L^ \gamma _tL^\rho _x } \lesssim \ell ^{-1} \lambda ^Nr_{\perp }^{\frac{1}{\rho }-\frac{1}{2}}r_{\parallel }^{\frac{1}{\rho }-\frac{1}{2}}\tau ^{\frac{1}{2}-\frac{1}{ \gamma }},\\&\Vert {\nabla }^N w_{q+1}^{(c)} \Vert _{L^\gamma _tL^\rho _x } +\Vert {\nabla }^N d_{q+1}^{(c)} \Vert _{L^\gamma _tL^\rho _x } \lesssim \ell ^{-1}\lambda ^Nr_{\perp }^{\frac{1}{\rho }+\frac{1}{2}}r_{\parallel }^{\frac{1}{\rho }-\frac{3}{2}}\tau ^{\frac{1}{2}-\frac{1}{\gamma }}, \\&\Vert  {\nabla }^Nw_{q+1}^{(t)} \Vert _{L^\gamma _tL^\rho _x }+ \Vert {\nabla }^N d_{q+1}^{(t)} \Vert _{L^\gamma _tL^\rho _x }\lesssim \ell ^{-2}\lambda ^N\mu ^{-1}r_{\perp }^{\frac{1}{\rho }-1}r_{\parallel }^{\frac{1}{\rho }-1}\tau ^{1-\frac{1}{\gamma }} ,\\&\Vert {\nabla }^N w_{q+1}^{(o)}\Vert _{L^\gamma _tL^\rho _x }\lesssim \ell ^{-8N-10}\sigma ^{-1},\ \ \Vert {\nabla }^N d_{q+1}^{(o)}\Vert _{L^\gamma _tL^\rho _x }\lesssim \ell ^{-4N-6}\sigma ^{-1} ,\\& \Vert  w_{q+1}^{(p)} \Vert _{C_{t,x}^N } + \Vert  w_{q+1}^{(c)} \Vert _{C_{t,x}^N }+\Vert  w_{q+1}^{(t)} \Vert _{C_{t,x}^N }+\Vert  w_{q+1}^{(o)}\Vert _{C_{t,x}^N }\lesssim \lambda ^{\frac{5N}{2}+3},\\&\Vert  d_{q+1}^{(p)} \Vert _{C_{t,x}^N } + \Vert  d_{q+1}^{(c)} \Vert _{C_{t,x}^N }+\Vert  d_{q+1}^{(t)} \Vert _{C_{t,x}^N }+\Vert  d_{q+1}^{(o)}\Vert _{C_{t,x}^N }\lesssim \lambda ^{\frac{5N}{2}+3}.\) 

By Lemmas REF , REF , REF  and REF 
and \(\ell ^{-6N_1}\lesssim \lambda ^{N_1}\) ,
for any \(\rho \in (1,+\infty )\) ,
\(& \Vert \nabla ^N w_{q+1}^{(p)} \Vert _{L^\gamma _tL^\rho _x } + \Vert \nabla ^N d_{q+1}^{(p)} \Vert _{L^\gamma _tL^\rho _x} \\\lesssim & \sum _{k \in \Lambda _u \cup \Lambda _B}\sum \limits _{N_1+N_2 = N}\Vert a_{(k)}\Vert _{C^{N_1}_{t,x}}\Vert g_{(\tau )}\Vert _{L_t^\gamma }\Vert  \nabla ^{N_2} W_{(k)} \Vert _{C_tL^\rho _x } \\& + \sum _{k \in \Lambda _B}\sum \limits _{N_1+N_2 = N}\Vert a_{(k)}\Vert _{C^{N_1}_{t,x}}\Vert g_{(\tau )}\Vert _{L_t^\gamma } \Vert \nabla ^{N_2} D_{(k)} \Vert _{C_tL^\rho _x} \\\lesssim & \ell ^{-1}\lambda ^Nr_{\perp }^{\frac{1}{\rho }-\frac{1}{2}}r_{\parallel }^{\frac{1}{\rho }-\frac{1}{2}}\tau ^{\frac{1}{2}-\frac{1}{\gamma }},\) 

which verifies (REF ).
Moreover, using (REF ), (REF ), (REF ) and Lemmas REF  and REF 
we have
\( & \quad \Vert {\nabla }^N w_{q+1}^{(c)} \Vert _{L^\gamma _tL^\rho _x} \\\lesssim & \sum _{k\in \Lambda _u \cup \Lambda _B}\left\Vert g_{(\tau )}{\nabla }^N\left( {\mathrm {curl}}(\nabla a_{(k)} \times W^c_{(k)})+ \nabla a_{(k)} \times {\mathrm {curl}}W_{(k)}^c+a_{(k)}\widetilde{W}^c_{(k)}\right) \right\Vert _{L^\gamma _tL^\rho _x} \\\lesssim &\sum \limits _{k\in \Lambda _u \cup \Lambda _B}\Vert g_{(\tau )}\Vert _{L^\gamma _t} \sum _{N_1+N_2=N}\left( \Vert  a_{(k)} \Vert _{C_{t,x}^{N_1+2}} \Vert {\nabla }^{N_2} W^c_{(k)}\Vert _{C_tW^{1,\rho }_x }+ \Vert  a_{(k)} \Vert _{C_{t,x}^{N_1}} \Vert  {\nabla }^{N_2}\widetilde{W}^c_{(k)}\Vert _{C_tL^\rho _x } \right) \nonumber \\\lesssim & \sum _{N_1+N_2=N} ( \ell ^{-8N_1-17}r_{\perp }^{\frac{1}{\rho } - \frac{1}{2}} r_{\parallel }^{\frac{1}{\rho } - \frac{1}{2}}\lambda ^{N_2-1}+ \ell ^{-8N_1-1}\lambda ^{N_2}r_{\perp }^{\frac{1}{\rho } + \frac{1}{2}} r_{\parallel }^{\frac{1}{\rho } - \frac{3}{2}})\tau ^{\frac{1}{2}-\frac{1}{\gamma }} \\\lesssim & (\ell ^{-17}r_{\perp }^{\frac{1}{\rho } - \frac{1}{2}} r_{\parallel }^{\frac{1}{\rho }- \frac{1}{2}}\lambda ^{N-1}+\ell ^{-1}r_{\perp }^{\frac{1}{\rho }+ \frac{1}{2}} r_{\parallel }^{\frac{1}{\rho } - \frac{3}{2}}\lambda ^{N})\tau ^{\frac{1}{2}-\frac{1}{\gamma }} \\\lesssim & \ell ^{-1}\lambda ^{N}r_{\perp }^{\frac{1}{\rho } + \frac{1}{2}} r_{\parallel }^{\frac{1}{\rho } - \frac{3}{2}}\tau ^{\frac{1}{2}-\frac{1}{\gamma }}.\) 

Similarly, by (), () and (),
\( &\quad \ \Vert {\nabla }^N d_{q+1}^{(c)} \Vert _{L^\gamma _tL^\rho _x} \\&\le \sum _{ k \in \Lambda _B} \left\Vert g_{(\tau )}{\nabla }^N \left( {\mathrm {curl}}(\nabla a_{(k)} \times D^c_{(k)})+ \nabla a_{(k)} \times {\mathrm {curl}}D^c_{(k)}+ a_{(k)}\widetilde{D}_{(k)}^c\right) \right\Vert _{L^\gamma _tL^\rho _x} \nonumber \\&\lesssim \sum \limits _{k\in \Lambda _B} \Vert g_{(\tau )}\Vert _{L^\gamma _t}\sum _{N_1+N_2=N} (\Vert  a_{(k)} \Vert _{C_{t,x}^{N_1+2}} \Vert {\nabla }^{N_2} D^c_{(k)}\Vert _{C_tW^{1,\rho }_x }+ \Vert a_{(k)}\Vert _{C^{N_1}_{t,x}} \Vert {\nabla }^{N_2}\widetilde{D}^c_{(k)}\Vert _{C_tL^\rho _x} ) \nonumber \\&\lesssim \sum _{N_1+N_2=N} (\ell ^{-4N_1-9} \lambda ^{N_2-1} r_{\perp }^{\frac{1}{\rho }- \frac{1}{2}} r_{\parallel }^{\frac{1}{\rho }- \frac{1}{2}}+ \ell ^{-4N_1-1} \lambda ^{N_2} r_{\perp }^{\frac{1}{\rho }+\frac{1}{2}} r_{\parallel }^{\frac{1}{\rho }-\frac{3}{2}} ) \tau ^{\frac{1}{2}-\frac{1}{\gamma }} \nonumber \\& \lesssim \ell ^{-1} \lambda ^{N} r_{\perp }^{\frac{1}{\rho }+ \frac{1}{2}} r_{\parallel }^{\frac{1}{\rho }- \frac{3}{2}} \tau ^{\frac{1}{2}-\frac{1}{\gamma }}.\) 

Thus, we obtain ().
Regarding the temporal correctors,
by (REF ), Lemmas REF , REF , REF  and REF 
and the boundedness of operators \(P_{\ne 0}\)  and \(\mathbb {P}_H\)  in \(L^\rho \) ,
\(&\Vert {\nabla }^N w_{q+1}^{(t)} \Vert _{L^\gamma _tL^\rho _x} \\\lesssim & \,\mu ^{-1} \sum _{k \in \Lambda _u \cup \Lambda _B}\Vert  g_{(\tau )}\Vert _{L^{2\gamma }_t }^2\sum _{N_1+N_2+N_3=N}\Vert \nabla ^{N_1}(a_{(k)}^2)\Vert _{C_{t,x} }\Vert  {\nabla }^{N_2} (\psi _{(k_1)}^2) \Vert _{C_tL^{\rho }_x }\Vert  {\nabla }^{N_3}(\phi _{(k)}^2) \Vert _{L^{\rho }_x } \\\lesssim & \,\mu ^{-1} \tau ^{1-\frac{1}{\gamma }}\sum _{N_1+N_2+N_3=N} \ell ^{-8N_1-2} \lambda ^{N_2} r_{\parallel }^{\frac{1}{\rho } -1}\lambda ^{N_3} r_{\perp }^{\frac{1}{\rho }-1} \\\lesssim & \, \ell ^{-2} \lambda ^{N}\mu ^{-1}r_{\perp }^{\frac{1}{\rho }-1} r_{\parallel }^{\frac{1}{\rho } -1}\tau ^{1-\frac{1}{\gamma }},\) 

and similarly,
\(\begin{aligned}\Vert {\nabla }^N d_{q+1}^{(t)} \Vert _{L^\gamma _tL^\rho _x}&\lesssim \ell ^{-2} \lambda ^{N}\mu ^{-1}r_{\perp }^{\frac{1}{\rho }-1} r_{\parallel }^{\frac{1}{\rho } -1}\tau ^{1-\frac{1}{\gamma }},\end{aligned}\) 
which yields ().
Concerning the estimates of temporal correctors \(w_{q+1}^{(o)}\)  and \(d_{q+1}^{(o)}\) ,
by (REF ), Lemmas REF  and REF ,
\(\begin{aligned}\Vert  {\nabla }^N d_{q+1}^{(o)}\Vert _{L^\gamma _tL^\rho _x }\lesssim \sigma ^{-1}\sum _{k \in \Lambda _B}\Vert h_{(\tau )}\Vert _{C_{t}} \Vert \nabla ^{N+1} (a^2_{(k)})\Vert _{C_{t,x}}\lesssim \ell ^{-4N-6} \sigma ^{-1},\end{aligned}\) 
and
\(\begin{aligned}\Vert  {\nabla }^Nw_{q+1}^{(o)}\Vert _{L^\gamma _tL^\rho _x} \lesssim \ell ^{-8N-10} \sigma ^{-1}.\end{aligned}\) 
It remains to prove the \(C^N\) -estimates () and () of perturbations.
By
Lemmas REF , REF , REF  and REF ,
\( \Vert  w_{q+1}^{(p)} \Vert _{C_{t,x}^N }\lesssim & \sum _{k \in \Lambda _u \cup \Lambda _B}\Vert a_{(k)}\Vert _{C_{t,x}^N }\sum _{0\le N_1+N_2 \le N} \Vert  g_{(\tau )}\Vert _{C_{t}^{N_1}}\Vert  W_{(k)} \Vert _{C_{t,x}^{N_2}} \\\lesssim & \sum _{0\le N_1+N_2 \le N} \sum _{ N_{21}+N_{22} = N_2}\ell ^{-8N-1} \sigma ^{N_1} \tau ^{N_1+\frac{1}{2}} r_{\perp }^{-\frac{1}{2}} r_{\parallel }^{-\frac{1}{2}} \lambda ^{N_{21}} \left(\frac{r_{\perp }\lambda \mu }{r_{\parallel }}\right)^{N_{22}} \\\lesssim & \lambda ^{\frac{5N+3}{2}},\) 

where we also used (REF ) and (REF ) in the last step.
Similarly, we have
\( &\quad \Vert  w_{q+1}^{(c)} \Vert _{C_{t,x}^N } \\& \lesssim \sum _{k\in \Lambda _u \cup \Lambda _B}\left\Vert g_{(\tau )}\left( {\mathrm {curl}}(\nabla a_{(k)} \times W^c_{(k)})+ \nabla a_{(k)} \times {\mathrm {curl}}W_{(k)}^c+a_{(k)}\widetilde{W}^c_{(k)}\right) \right\Vert _{C_{t,x}^N } \\& \lesssim \sum _{k \in \Lambda _u \cup \Lambda _B}\Vert a_{(k)}\Vert _{C_{t,x}^{N+2}}\sum _{0\le N_1+N_2 \le N} \Vert  g_{(\tau )}\Vert _{C_{t}^{N_1}}\left(\Vert  W^c_{(k)} \Vert _{C_{t,x}^{N_2}} + \Vert  \nabla W^c_{(k)} \Vert _{C_{t,x}^{N_2}}+ \Vert \widetilde{W}^c_{(k)}\Vert _{C^{N_2}_{t,x}} \right) \nonumber \\& \lesssim \sum _{0\le N_1+N_2 \le N}\ell ^{-8N-1} \sigma ^{N_1} \tau ^{N_1+\frac{1}{2}} r_{\perp }^{-\frac{1}{2}} r_{\parallel }^{-\frac{1}{2}} \left(\frac{r_{\perp }\lambda \mu }{r_{\parallel }}\right)^{N_2}\left(\ell ^{-16}\lambda ^{-1} + \ell ^{-16} \frac{r_{\perp }}{r_{\parallel }}\right) \\& \lesssim \lambda ^{\frac{5N+3}{2}}.\) 

Moreover, by Sobolev's embedding \(W^{1,6}(\mathbb {T}^3) \hookrightarrow L^{\infty }(\mathbb {T}^3)\) 
and the boudedness of operators \(\mathbb {P}_H \mathbb {P}_{\ne 0}\)  in the space \(L^6_x\) ,
\( \Vert  w_{q+1}^{(t)} \Vert _{C_{t,x}^N }& \lesssim \mu ^{-1} \sum _{k \in \Lambda _u \cup \Lambda _B}\Vert a_{(k)}^2 g_{(\tau )}^2 \psi ^2_{(k_1)} \phi ^2_{(k)}\Vert _{C^N_t W^{N+1,6}_x} \\& \lesssim \mu ^{-1} \sum _{k \in \Lambda _u \cup \Lambda _B}\sum _{0\le N_1+N_2 \le N+1} \Vert a_{(k)}^2\Vert _{C_{t,x}^N }\Vert g_{(\tau )}^2\psi ^2_{(k_1)} \phi ^2_{(k)}\Vert _{C_{t}^{N_1} C^{N_2}_x} \\& \le \frac{1}{3} \lambda ^{\frac{5N}{2}+3},\) 

where we also used Lemmas REF , REF , REF  and REF  in the last step.
Finally, arguing as above we get
\( \Vert  w_{q+1}^{(o)}\Vert _{C_{t,x}^N }& \lesssim \sigma ^{-1} \sum _{k \in \Lambda _u \cup \Lambda _B} \Vert h_{(\tau )} {\nabla }(a_{(k)}^2) \Vert _{C^N_tW^{N+1,6}_x} \\& \lesssim \sigma ^{-1} \sum _{k \in \Lambda _u \cup \Lambda _B} \Vert h_{(\tau )}\Vert _{C_{t}^{N+1}} \Vert \nabla (a^2_{(k)})\Vert _{C_{t,x}^{N+1}} \\& \lesssim \sigma ^N \tau ^{N+1} \ell ^{-8(N+1)-2} \\& \lesssim \lambda ^{2{\varepsilon }N} \lambda ^{(1-8{\varepsilon })(N+1)-2} \lambda ^{\frac{84}{100}{\varepsilon }}\lesssim \lambda ^{N+1},\) 

where the last two steps were due to (REF ) and (REF ).
Thus, combining (REF )-(REF ) altogether and using (REF )
we conclude that
\( & \Vert  w_{q+1}^{(p)} \Vert _{C_{t,x}^N } + \Vert  w_{q+1}^{(c)} \Vert _{C_{t,x}^N }+\Vert  w_{q+1}^{(t)} \Vert _{C_{t,x}^N }+\Vert  w_{q+1}^{(o)}\Vert _{C_{t,x}^N }\le \frac{1}{2} \lambda ^{\frac{5N}{2}+3}.\) 

The \(C^N\) -estimates of the magnetic perturbations in () can be proved similarly.

Verification of inductive estimates (REF ), (REF ) and (REF )
We are now in position to verify the inductive estimates (REF ), (REF ) and (REF )
for the velocity and magnetic fields.
Note that,
the previous estimate (REF ) cannot yield the decay properties required by the
inductive estimates (REF ) and (REF ).
An important ingredient here is the \(L^p\)  decorrelation Lemma REF  below,
which permits to derive the decay of the \(L^2_{t,x}\) -norm of principal parts.
Lemma 4.7 ([20]}, Lemma 2.4; see also [11]}, Lemma 3.7)  
Let \(\sigma \in \mathbb {N}\)  and \(f,g:\mathbb {T}^d\rightarrow {\mathbb {R}}\)  be smooth functions. Then for every \(p\in [1,+{\infty }]\) ,
\(\big |\Vert fg(\sigma \cdot )\Vert _{L^p(d)}-\Vert f\Vert _{L^p(d)}\Vert g\Vert _{L^p(d)} \big |\lesssim \sigma ^{-\frac{1}{p}}\Vert f\Vert _{C^1(d)}\Vert g\Vert _{L^p(d)}.\) 

We apply Lemma REF  with \(f= a_{(k)}\) , \(g = g_{(\tau )}\psi _{(k_1)}\phi _{(k)}\)  and \(\sigma = \lambda ^{2{\varepsilon }}\) .
By (REF ), (REF ) and Lemmas REF  and REF ,
\(\Vert w^{(p)}_{q+1}\Vert _{L^2_{t,x}}&\lesssim \sum \limits _{k\in \Lambda _u \cup \Lambda _B}\Big (\Vert a_{(k)}\Vert _{L^2_{t,x}}\Vert  g_{(\tau )}\Vert _{L^2_{t}} \Vert  \psi _{(k_1)}\phi _{(k)}\Vert _{C_tL^2_{x}} \\&\qquad \qquad \qquad +\sigma ^{-\frac{1}{2}}\Vert a_{(k)}\Vert _{C^1_{t,x}}\Vert  g_{(\tau )}\Vert _{L^2_{t}} \Vert  \psi _{(k_1)}\phi _{(k)}\Vert _{C_tL^2_{x}}\Big ) \\&\lesssim \delta _{q+1}^{\frac{1}{2}}+\ell ^{-8}\lambda ^{-{\varepsilon }}_{q+1} \lesssim \delta _{q+1}^{\frac{1}{2}}, \\\Vert d^{(p)}_{q+1}\Vert _{L^2_{t,x}}&\lesssim \sum \limits _{k \in \Lambda _B}\Big (\Vert a_{(k)}\Vert _{L^2_{t,x}}\Vert  g_{(\tau )}\Vert _{L^2_{t}} \Vert  \psi _{(k_1)}\phi _{(k)}\Vert _{C_tL^2_{x}}\\&\qquad \qquad \quad +\sigma ^{-\frac{1}{2}}\Vert a_{(k)}\Vert _{C^1_{t,x}}\Vert  g_{(\tau )}\Vert _{L^2_{t}} \Vert  \psi _{(k_1)}\phi _{(k)}\Vert _{C_tL^2_{x}} \Big ) \\&\lesssim \delta _{q+1}^{\frac{1}{2}}+\ell ^{-4}\lambda ^{-{\varepsilon }}_{q+1} \lesssim \delta _{q+1}^{\frac{1}{2}}.\) 
Below we verify the iterative estimates for \(u_{q+1}\)  and \(B_{q+1}\) .
For the velocity perturbation \(w_{q+1}\) ,
using (REF ), (REF ) and Lemma REF 
we have
\( \Vert w_{q+1}\Vert _{L^2_{t,x}} &\lesssim \Vert w_{q+1}^{(p)} \Vert _{L^2_{t,x}} + \Vert  w_{q+1}^{(c)} \Vert _{L^2_{t,x}} +\Vert  w_{q+1}^{(t)} \Vert _{L^2_{t,x}}+\Vert  w_{q+1}^{(o)}\Vert _{L^2_{t,x}}\\&\lesssim \delta _{q+1}^{\frac{1}{2}} +\ell ^{-1}r_{\perp } r_{\parallel }^{-1}+ \ell ^{-2} \mu ^{-1} r_{\perp }^{-\frac{1}{2}} r_{\parallel }^{-\frac{1}{2}} \tau ^\frac{1}{2} + \ell ^{-10}\sigma ^{-1}\lesssim \delta _{q+1}^{\frac{1}{2}},\) 

and
\( \Vert w_{q+1}\Vert _{L^1_tL^2_x} &\lesssim \Vert w_{q+1}^{(p)} \Vert _{L^1_tL^2_x} + \Vert  w_{q+1}^{(c)} \Vert _{L^1_tL^2_x} +\Vert  w_{q+1}^{(t)} \Vert _{L^1_tL^2_x}+\Vert  w_{q+1}^{(o)}\Vert _{L^1_tL^2_x}\\&\lesssim \ell ^{-1}\tau ^{-\frac{1}{2}}+\ell ^{-1} r_{\perp } r_{\parallel }^{-1} \tau ^{-\frac{1}{2}}+ \ell ^{-2} \mu ^{-1} r_{\perp }^{-\frac{1}{2}} r_{\parallel }^{-\frac{1}{2}} + \ell ^{-10}\sigma ^{-1}\lesssim \lambda _{q+1}^{-{\varepsilon }} ,\) 

where we also used (REF ) in the last steps.
Then, in view of (REF ), (),
(REF ) and the standard mollification estimates,
we get for \(a\)  sufficiently large,
\(& \Vert u_{q+1}\Vert _{C^1_{t,x}} \lesssim \Vert u_{\ell }\Vert _{C^1_{t,x}}+\Vert w_{q+1}\Vert _{C^1_{t,x}}\lesssim \lambda _{q}^7+ \lambda _{q+1}^{\frac{13}{2}}\le \lambda _{q+1}^7,  \\& \Vert u_{q} - u_{q+1}\Vert _{L^2_{t,x}} \le \Vert  u_{q} - u_{\ell } \Vert _{L^2_{t,x}} + \Vert u_{\ell } - u_{q+1}\Vert _{L^2_{t,x}} \nonumber \\&\qquad \qquad \qquad \ \lesssim \Vert  u_q - u_{\ell } \Vert _{C_{t,x}}+ \Vert w_{q+1}\Vert _{L^2_{t,x}} \nonumber \\&\qquad \qquad \qquad \ \lesssim \ell \Vert u_q \Vert _{C_{t,x}^1}+\delta _{q+1}^{\frac{1}{2}} \le M\delta _{q+1}^{\frac{1}{2}}, \) 

and
\( \Vert u_{q} - u_{q+1}\Vert _{L^1_tL^2_x} \lesssim & \Vert  u_q - u_{\ell } \Vert _{C_{t,x}}+ \Vert w_{q+1}\Vert _{L^1_tL^2_x} \nonumber \\\lesssim & \ell \lambda _q^7+\lambda _{q+1}^{-{\varepsilon }} \le \delta _{q+2}^{\frac{1}{2}},\) 

where we also used the inequalities
\(\lambda _q^{-13} \ll \delta _{q+1}^{\frac{1}{2}}\) 
and \(\ell \lambda _q^7 \ll \delta _{q+2}^{\frac{1}{2}}\) ,
due to (REF ) and \(M\)  is a fixed large universal constant.
Similarly, for the magnetic perturbation \(d_{q+1}\) ,
by (), () and Lemma REF ,
\( \Vert d_{q+1} \Vert _{L^2_{t,x}} & \le \Vert  d_{q+1}^{(p)}\Vert _{L^2_{t,x}} +\Vert  d_{q+1}^{(c)} \Vert _{L^2_{t,x}}+\Vert  d_{q+1}^{(t)} \Vert _{L^2_{t,x}}+\Vert  d_{q+1}^{(o)}\Vert _{L^2_{t,x}} \nonumber \\& \lesssim \delta _{q+1}^{\frac{1}{2}} +\ell ^{-1}r_{\perp } r_{\parallel }^{-1}+ \ell ^{-2} \mu ^{-1}r_{\perp }^{-\frac{1}{2}} r_{\parallel }^{-\frac{1}{2}}\tau ^{\frac{1}{2}} +\ell ^{-6}\sigma ^{-1} \lesssim \delta _{q+1}^{\frac{1}{2}},\) 

and
\( \Vert d_{q+1}\Vert _{L^1_tL^2_x} &\lesssim \Vert d_{q+1}^{(p)} \Vert _{L^1_tL^2_x} + \Vert  d_{q+1}^{(c)} \Vert _{L^1_tL^2_x} +\Vert d_{q+1}^{(t)} \Vert _{L^1_tL^2_x}+ \Vert d_{q+1}^{(o)}\Vert _{L^1_tL^2_x}\\&\lesssim \ell ^{-1}\tau ^{-\frac{1}{2}}+\ell ^{-1} r_{\perp } r_{\parallel }^{-1} \tau ^{-\frac{1}{2}}+ \ell ^{-2} \mu ^{-1} r_{\perp }^{-\frac{1}{2}} r_{\parallel }^{-\frac{1}{2}} + \ell ^{-6}\sigma ^{-1}\le \lambda _{q+1}^{-{\varepsilon }} ,\) 

which along with (REF ), () and Lemma REF  yield that
\(\Vert B_{q+1}\Vert _{C^1_{t,x}}& \lesssim \Vert B_{\ell }\Vert _{C^1_{t,x}}+\Vert d_{q+1}\Vert _{C^1_{t,x}}\lesssim \lambda _{q}^7+ \lambda _{q+1}^{\frac{13}{2}}\le \lambda _{q+1}^7,  \\\Vert B_{q} - B_{q+1} \Vert _{L^2_{t,x}}& \lesssim \Vert  B_q - B_{\ell } \Vert _{C_{t,x}}+ \Vert d_{q+1}\Vert _{L^2_{t,x}} \nonumber \\&\lesssim \ell \Vert B_q \Vert _{C_{t,x}^1}+ \delta _{q+1}^{\frac{1}{2}} \le M\delta _{q+1}^{\frac{1}{2}}, \) 

and
\( \Vert B_{q} - B_{q+1}\Vert _{L^1_tL^2_x} & \lesssim \Vert  B_q - B_{\ell } \Vert _{C_{t,x}}+ \Vert d_{q+1}\Vert _{L^1_tL^2_x}\lesssim \ell \lambda _q^7+\lambda _{q+1}^{-{\varepsilon }} \le \delta _{q+2}^{\frac{1}{2}}.\) 

Therefore, the inductive estimates in (REF ), (REF ) and (REF ) are verified.

Reynolds and magnetic stresses
The main purpose of this section is to determine the Reynolds and magnetic stresses
and to prove the main iteration in Theorem REF .
The important roles here are played by the inverse-divergence operators \(\mathcal {R}^u\)  and \(\mathcal {R}^B\) , defined by
\(& (\mathcal {R}^u v)^{kl} := \partial _k \Delta ^{-1} v^l + \partial _l \Delta ^{-1} v^k - \frac{1}{2}(\delta _{kl} + \partial _k \partial _l \Delta ^{-1}){\mathrm {div}}\Delta ^{-1} v, \\& (\mathcal {R}^Bf)_{ij} := \varepsilon _{ijk} (-\Delta )^{-1}({\mathrm {curl}}f)_k,\) 

where \(\int _{\mathbb {T}^3} v dx =0\) , \({\mathrm {div}}f=0\) ,
and \(\varepsilon _{ijk}\)  is the Levi-Civita tensor, \(i,j,k,l \in \lbrace 1,2,3\rbrace \) .
The operator \(\mathcal {R}^u\)  returns symmetric and trace-free matrices,
while the operator \(\mathcal {R}^B\)  returns skew-symmetric matrices.
Moreover, one has the algebraic identities
\({\mathrm {div}}\mathcal {R}^u(v) = v,\ \ {\mathrm {div}}\mathcal {R}^B(f) = f.\) 

Both
\(|\nabla |\mathcal {R}^u\)  and \(|\nabla |\mathcal {R}^B\)  are Calderon-Zygmund operators
and thus they are bounded in the spaces \(L^p\) , \(1<p<+\infty \) .
See [2]}, [42]} for more details.
Decomposition of magnetic stress
Using (REF ) with \(q+1\)  replacing \(q\) ,
(REF ), (REF ), and (REF )
we derive the equation for the magnetic stress \(\mathring{R}^B_{q+1}\) :
\(&\displaystyle {\mathrm {div}}\mathring{R}_{q+1}^B \\&=\underbrace{\partial _t (d^{(p)}_{q+ 1}+d^{(c)}_{q+ 1})+\nu _2(-\Delta )^{\alpha _2} d_{q+1} +{\mathrm {div}}(d_{q + 1} \otimes u_{\ell } - u_{\ell } \otimes d_{q+1}+ B_{\ell }\otimes w_{q+1} -w_{q +1} \otimes B_{\ell } )}_{ {\mathrm {div}}\mathring{R}_{lin}^B } \\&\quad +\underbrace{{\mathrm {div}}(d_{q+ 1}^{(p)} \otimes w_{q+1}^{(p)} -w_{q+ 1}^{(p)} \otimes d_{q+ 1}^{(p)} + \mathring{R}_{\ell }^B)+ \partial _t d_{q+1}^{(t)}+ \partial _t d_{q+1}^{(o)}}_{{\mathrm {div}}\mathring{R}_{osc}^B } \\&\quad +{\mathrm {div}}\Big ( d_{q+1}^{(p)} \otimes (w_{q+1}^{(c)}+ w_{q+1}^{(t)}+w_{q+1}^{(o)}) -(w_{q+1}^{(c)}+w_{q+1}^{(t)}+w_{q+1}^{(o)}) \otimes d_{q+1} \\&\qquad \underbrace{\qquad \quad +(d_{q+1}^{(c)}+d_{q+1}^{(t)}+d_{q+1}^{(o)})\otimes w_{q+1}-w_{q+1}^{(p)} \otimes (d_{q+1}^{(c)}+d_{q+1}^{(t)}+d_{q+1}^{(o)})\Big ) }_{{\mathrm {div}}\mathring{R}_{cor}^B } \\&\quad +\underbrace{ {\mathrm {div}}\left(B_{\ell } \otimes u_{\ell }-u_{\ell }\otimes B_{\ell }-\left({u_q}-{u_q}\otimes ) *_{x} \phi _{\ell } *_{t} \varphi _{\ell }\right)\right._{{\mathrm {div}}\mathring{R}_{com}^B } . }\) 
Using the inverse divergence operator \(\mathcal {R}^B\) 
we define the magnetic stress by
\(\mathring{R}_{q+1}^B := \mathring{R}_{lin}^B + \mathring{R}_{osc}^B+ \mathring{R}_{cor}^B+\mathring{R}_{com}^B,\) 

where
\(\mathring{R}_{lin}^B:= & \mathcal {R}^B\left(\partial _t (d^{(p)}_{q+1}+d^{(c)}_{q+1})\right)+ \nu _2 \mathcal {R}^B (-\Delta )^{{\alpha }_2} d_{q+1} \nonumber \\& +\mathcal {R}^B\mathbb {P}_{H}{\mathrm {div}}\left( d_{q + 1} \otimes u_{\ell } - u_{\ell } \otimes d_{q+1}+ B_{\ell }\otimes w_{q+1} -w_{q +1} \otimes B_{\ell }\right),\) 

the oscillation error
\( \mathring{R}_{osc}^B &:= \sum _{k \in \Lambda _B}\mathcal {R}^B\mathbb {P}_{H}\mathbb {P}_{\ne 0}\left(g_{(\tau )}^2 \mathbb {P}_{\ne 0}(D_{(k)}\otimes W_{(k)}-W_{(k)}\otimes D_{(k)} )\nabla (a_{(k)}^2)\right)\\&\quad -\mu ^{-1} \sum _{k \in \Lambda _B}\mathcal {R}^B\mathbb {P}_{H}\mathbb {P}_{\ne 0}(\partial _t (a_{(k)}^2g_{(\tau )}^2)\psi _{(k_1)}^2\phi _{(k)}^2k_2)\\&\quad -\sigma ^{-1}\sum _{k\in \Lambda _B}\mathcal {R}^B\mathbb {P}_{H}\mathbb {P}_{\ne 0}\left(h_{(\tau )}{}_{3}D_{(k)}\otimes W_{(k)}-W_{(k)}\otimes D_{(k)} {\rm d}x\partial _t\nabla (a_{(k)}^{2})\right),\) 

the corrector error
\( \mathring{R}_{cor}^B := &\,\mathcal {R}^B\mathbb {P}_{H}{\mathrm {div}}\bigg ( d_{q+1}^{(p)} \otimes (w_{q+1}^{(c)}+ w_{q+1}^{(t)}+w_{q+1}^{(o)}) -(w_{q+1}^{(c)}+w_{q+1}^{(t)}+w_{q+1}^{(o)}) \otimes d_{q+1}\\&\qquad \qquad + (d_{q+1}^{(c)}+d_{q+1}^{(t)}+d_{q+1}^{(o)})\otimes w_{q+1} -w_{q+1}^{(p)} \otimes (d_{q+1}^{(c)}+d_{q+1}^{(t)}+d_{q+1}^{(o)}) \bigg ),\) 

and the commutator error
\( \mathring{R}_{com}^B= \mathcal {R}^u \mathbb {P}_H {\mathrm {div}}\left(B_{\ell } \otimes u_{\ell }-u_{\ell }\otimes B_{\ell }-\left({u_q}-{u_q}\otimes ) *_{x} \phi _{\ell } *_{t} \varphi _{\ell }\right).\right.\) 
Remark 5.1 Since
\( \mathbb {P}_H \mathbb {P}_{\ne 0} =\mathbb {P}_{\ne 0} \mathbb {P}_H,\ \ \mathbb {P}_H {\mathrm {div}}= \mathbb {P}_{\ne 0} \mathbb {P}_H {\mathrm {div}},\) 

by (REF ) and
the definitions () and (),
\( {\mathrm {div}}(d^{(p)}_{q+1}+ d^{(c)}_{q+1})=0, \ \ {\mathrm {div}}(-\Delta )^{{\alpha }_2} d_{q+1} =0.\) 

Taking into account \({\mathrm {div}}\mathbb {P}_H =0\)  we see that
the terms in (REF )-(REF ) are
in the domain of the operator \(\mathcal {R}^B\) .

By (REF ),
\( d^{(p)}_{q+1} + d^{(c)}_{q+1} = \mathbb {P}_H (d^{(p)}_{q+1} + d^{(c)}_{q+1}), \ \ (-\Delta )^{{\alpha }_2} d_{q+1} = \mathbb {P}_H (-\Delta )^{{\alpha }_2} d_{q+1}.\) 
We also note that,
in contrast to the nonlinearities in the velocity equation,
the nonlinear terms in the magnetic equation is skew symmetric,
which, in particular, yields that
\( {\mathrm {div}}({\mathrm {div}}\mathring{R}^B_{q+1}) =0.\) 
Thus, by virtue of (REF ) and (REF ),
we obtain
\({\mathrm {div}}\mathring{R}^B_{q+1}= \mathbb {P}_H {\mathrm {div}}\mathring{R}^B_{q+1}.\) 
Then, using the algebraic identities
\({\mathrm {div}}\mathcal {R}^B = \mathrm {Id}\) ,
(REF ), (REF ) and (REF ),
together with the facts that
\({\mathrm {div}}= {\mathrm {div}}\mathbb {P}_{\ne 0} = \mathbb {P}_{\ne 0} {\mathrm {div}}\) ,
we obtain that
\(\mathring{R}^B_{q+1} \)  satisfies the relaxation magnetic equation in (REF ) and
the following identity holds
\( \mathring{R}^B_{q+1} = \mathcal {R}^B \mathbb {P}_H {\mathrm {div}}\mathring{R}^B_{q+1} .\) 

Decomposition of Reynolds stress
Concerning the Reynolds stress we compute
\(&\displaystyle {\mathrm {div}}\mathring{R}_{q+1}^u - \nabla P_{q+1} \\&\displaystyle = \underbrace{\partial _t (w_{q+1}^{(p)}+w_{q+1}^{(c)}) +\nu _1(-\Delta )^{\alpha _1} w_{q+1} +{\mathrm {div}}\big (u_{\ell } \otimes w_{q+1} + w_{q+ 1} \otimes u_{\ell } - B_{\ell } \otimes d_{q+1} - d_{q+1} \otimes B_{\ell }\big ) }_{ {\mathrm {div}}\mathring{R}_{lin}^u +\nabla P_{lin} } \\&\displaystyle \quad + \underbrace{{\mathrm {div}}(w_{q+1}^{(p)} \otimes w_{q+1}^{(p)} - d_{q+1}^{(p)} \otimes d_{q+1}^{(p)} + \mathring{R}_{\ell }^u)+\partial _t w_{q+1}^{(t)}+\partial _t w_{q+1}^{(o)}}_{{\mathrm {div}}\mathring{R}_{osc}^u +\nabla P_{osc}} \\&\displaystyle \quad +{\mathrm {div}}\Big ((w_{q+1}^{(c)}+ w_{q+1}^{(t)}+w_{q+1}^{(o)})\otimes w_{q+1}+ w_{q+1}^{(p)} \otimes (w_{q+1}^{(c)}+ w_{q+1}^{(t)}+w_{q+1}^{(o)}) \\&\qquad \underbrace{\qquad - (d_{q+1}^{(c)}+ d_{q+1}^{(t)}+d_{q+1}^{(o)})\otimes d_{q+1}- d_{q+1}^{(p)} \otimes (d_{q+1}^{(c)}+ d_{q+1}^{(t)}+d_{q+1}^{(o)}) \Big )}_{{\mathrm {div}}\mathring{R}_{cor}^u +\nabla P_{cor}}\\&\displaystyle \quad + \underbrace{ {\mathrm {div}}\left(u_{\ell }\mathring{\otimes } u_{\ell }-B_{\ell } \mathring{\otimes }B_{\ell }-\left({u_q}\mathring{\otimes }{u_q}-{\otimes }) *_{x} \phi _{\ell } *_{t} \varphi _{\ell }\right)\right._{{\mathrm {div}}\mathring{R}_{com}^u} -\nabla P_{\ell }. }\) 
Then, using the inverse divergence operator \(\mathcal {R}^u\) 
we define the Reynolds stress by
\(\mathring{R}_{q+1}^u := \mathring{R}_{lin}^u + \mathring{R}_{osc}^u+ \mathring{R}_{cor}^u+\mathring{R}_{com}^u,\) 

where
\(\mathring{R}_{lin}^u & := \mathcal {R}^u\left(\partial _t (w_{q+1}^{(p)} +w_{q+1}^{(c)} )\right)+ \nu _1 \mathcal {R}^u (-\Delta )^{{\alpha }_1} w_{q+1} \nonumber \\&\quad + \mathcal {R}^u\mathbb {P}_H {\mathrm {div}}\left(u_{\ell } \mathring{\otimes } w_{q+1} + w_{q+ 1}\mathring{\otimes } u_{\ell }- B_{\ell } \mathring{\otimes } d_{q+1} - d_{q+1} \mathring{\otimes } B_{\ell }\right), \) 

the oscillation error
\(\mathring{R}_{osc}^u &:= \sum _{k \in \Lambda _u} \mathcal {R}^u \mathbb {P}_H\mathbb {P}_{\ne 0}\left(g_{(\tau )}^2 \mathbb {P}_{\ne 0}(W_{(k)}\otimes W_{(k)})\nabla (a_{(k)}^2)\right) \\&\quad + \sum _{k \in \Lambda _B} \mathcal {R}^u \mathbb {P}_H\mathbb {P}_{\ne 0}\left(g_{(\tau )}^2 \mathbb {P}_{\ne 0}(W_{(k)}\otimes W_{(k)}-D_{(k)}\otimes D_{(k)})\nabla (a_{(k)}^2)\right)\\&\quad -\mu ^{-1}\sum _{k \in \Lambda _u\cup \Lambda _B}\mathcal {R}^u \mathbb {P}_H \mathbb {P}_{\ne 0}\left(\partial _t (a_{(k)}^2g_{(\tau )}^2)\psi _{(k_1)}^2\phi _{(k)}^2k_1\right)\\&\quad -\sigma ^{-1}\sum _{k\in \Lambda _u}\mathcal {R}^u \mathbb {P}_H \mathbb {P}_{\ne 0}\left(h_{(\tau )}{}_{3}W_{(k)}\otimes W_{(k)}{\rm d}x\partial _t\nabla (a_{(k)}^{2})\right)\\&\quad -\sigma ^{-1}\sum _{k\in \Lambda _B}\mathcal {R}^u \mathbb {P}_H \mathbb {P}_{\ne 0}\left( h_{(\tau )}{}_{3}W_{(k)}\otimes W_{(k)}-D_{(k)}\otimes D_{(k)}{\rm d}x\partial _t\nabla (a_{(k)}^{2})\right),\) 

the corrector error
\(\mathring{R}_{cor}^u &:= \mathcal {R}^u \mathbb {P}_H {\mathrm {div}}\bigg ( w^{(p)}_{q+1} \mathring{\otimes } (w_{q+1}^{(c)}+w_{q+1}^{(t)}+w_{q+1}^{(o)})+ (w_{q+1}^{(c)}+w_{q+1}^{(t)}+w_{q+1}^{(o)}) \mathring{\otimes } w_{q+1} \nonumber \\&\qquad \qquad \qquad - d^{(p)}_{q+1} \mathring{\otimes } (d_{q+1}^{(c)}+d_{q+1}^{(t)}+d_{q+1}^{(o)})- (d_{q+1}^{(c)}+d_{q+1}^{(t)}+d_{q+1}^{(o)}) \mathring{\otimes } d_{q+1} \bigg ), \) 

and the commutator error
\( \mathring{R}^u_{com}:=\mathcal {R}^u \mathbb {P}_H {\mathrm {div}}\left(B_{\ell } \mathring{\otimes }u_{\ell }-u_{\ell } \mathring{\otimes } B_{\ell }-({\otimes } {u_q}-{u_q}\mathring{\otimes }  *_{x} \phi _{\ell } *_{t} \varphi _{\ell }\right).\) 
Remark 5.2 We infer from (REF ) that
the above terms in (REF )-(REF ) are
in the domain of the operator \(\mathcal {R}^u\) .

As in (REF ), by (REF ),
the definitions (REF ) and (REF ),
and \({\mathrm {div}}\mathbb {P}_H=0\) ,
\( {\mathrm {div}}( w^{(p)}_{q+1} + w^{(c)}_{q+1} ) =0, \ \ {\mathrm {div}}( w^{(t)}_{q+1} + w^{(o)}_{q+1} ) =0,\) 

which yield that
\(w^{(p)}_{q+1} + w^{(c)}_{q+1} = \mathbb {P}_H (w^{(p)}_{q+1} + w^{(c)}_{q+1}), \ \ (-\Delta )^{{\alpha }_1} w_{q+1} = \mathbb {P}_H (-\Delta )^{{\alpha }_1} w_{q+1}.\) 

Then, using the algebraic identities \({\mathrm {div}}\mathcal {R}^u = \mathrm {Id}\) ,
(REF ), (REF ) and (REF )
we obtain
\({\mathrm {div}}\mathring{R}^u_{q+1}= \mathbb {P}_H \left( \partial _t {u_q}+\nu _1(-\Delta )^{\alpha _1} {u_q}+ {\mathrm {div}}({u_q}\otimes {u_q}-\right),\) 

which yields that \(\mathring{R}^u_{q+1}\)  satisfies the relaxation velocity equation in (REF ).
Moreover,
since \( \mathbb {P}^2_H = \mathbb {P}_H\) ,
it also holds that
\( \mathring{R}^u_{q+1} = \mathcal {R}^u \mathbb {P}_H {\mathrm {div}}\mathring{R}^u_{q+1}.\) 

Estimates of magnetic stress
The purpose of this subsection is to estimate the magnetic stress \(\mathring{R}^B_{q+1}\) 
given by (REF ) at level \(q+1\) .
Since the Calderón-Zygmund operators are bounded in the space \(L^p_x\)  for any \(1<p<+{\infty }\) ,
we choose
\(p: =\frac{2-8\varepsilon }{2-9\varepsilon }\in (1,2),\) 
where \({\varepsilon }\)  is given by (REF ).
In particular,
\((1-4\varepsilon )(1-\frac{1}{p})=\frac{\varepsilon }{2},\) 
which, via (REF ), yields that
\( r_{\perp }^{\frac{1}{p}-1}r_{\parallel }^{\frac{1}{p}-1} = \lambda ^{\varepsilon },\quad r_{\perp }^{\frac{1}{p}-\frac{1}{2}}r_{\parallel }^{\frac{1}{p}-\frac{1}{2}} = \lambda ^{-1+5\varepsilon }.\) 
Let us estimate the four parts in the decomposition of \(\mathring{R}^B_{q+1}\)  separately.
Linear error.
Note that,
by (),
\(& \Vert  \mathcal {R}^B\partial _t( d_{q+1}^{(p)}+ d_{q+1}^{(c)})\Vert _{L_t^1L_x^p} \nonumber \\\lesssim & \sum _{k \in \Lambda _B}\Vert  \mathcal {R}^B {\mathrm {curl}}{\mathrm {curl}}\partial _t(g_{(\tau )}a_{(k)} D^c_{(k)}) \Vert _{L_t^1L_x^p} \nonumber \\\lesssim & \sum _{k \in \Lambda _B}\Vert  {\mathrm {curl}}\partial _t (g_{(\tau )}a_{(k)} D^c_{(k)}) \Vert _{L_t^1L_x^p} \nonumber \\\lesssim & \sum _{k \in \Lambda _B}\Big (\Vert  g_{(\tau )}\Vert _{L^1_t}\left(\Vert  a_{(k)} \Vert _{C_{t,x}^2}\Vert  D^c_{(k)} \Vert _{C_t W_x^{1,p}}+\Vert  a_{(k)} \Vert _{C_{t,x}^1}\Vert  \partial _t D^c_{(k)} \Vert _{C_t W^{1,p}_x}\right) \nonumber \\&\qquad \qquad +\Vert  \partial _tg_{(\tau )}\Vert _{L_t^1}\Vert  a_{(k)} \Vert _{C_{t,x}^1}\Vert  D^c_{(k)} \Vert _{C_t W_x^{1,p}}\Big ).\) 

Then, by Lemmas REF  and REF ,
(REF ), (REF ) and (REF ),
\( & \Vert  \mathcal {R}^B\partial _t( d_{q+1}^{(p)}+ d_{q+1}^{(c)})\Vert _{L_t^1L_x^p} \\\lesssim & \tau ^{-\frac{1}{2}}(\ell ^{-8}r_{\perp }^{\frac{1}{p}-\frac{1}{2}}r_{\parallel }^{\frac{1}{p}-\frac{1}{2}}\lambda ^{-1}+\ell ^{-4}r_{\perp }^{\frac{1}{p}+\frac{1}{2}}r_{\parallel }^{\frac{1}{p}-\frac{3}{2}}\mu )+ \sigma \tau ^{\frac{1}{2}}\ell ^{-4}r_{\perp }^{\frac{1}{p}-\frac{1}{2}}r_{\parallel }^{\frac{1}{p}-\frac{1}{2}}\lambda ^{-1} \\\lesssim & \ell ^{-8}(\lambda ^{-\frac{3}{2}+4{\varepsilon }}+ \lambda ^{-2{\varepsilon }})\lesssim \ell ^{-8}\lambda ^{-2{\varepsilon }}.\) 
The control of the fractional viscosity \((-\Delta )^{{\alpha }_2}\) 
requires both the temporal and spatial intermittency of the magnetic flows.
More precisely, by Lemma REF ,
for \({\alpha }_2\in [0,1/2]\) ,
\( \Vert  \mathcal {R}^B(-\Delta )^{\alpha _2} d_{q+1} \Vert _{L_t^1L^p_x}\lesssim & \Vert d_{q+1}\Vert _{L_t^1L^p_x} \\\lesssim & \ell ^{-1} r_{\perp }^{\frac{1}{p} - \frac{1}{2}} r_{\parallel }^{\frac{1}{p} - \frac{1}{2}}\tau ^{-\frac{1}{2}}+\ell ^{-6}\sigma ^{-1} \\\lesssim & \ell ^{-6}\lambda ^{-2{\varepsilon }}\lesssim \ell ^{-1} \lambda ^{-{\varepsilon }}.\) 

For the stronger viscosity \((-\Delta )^{{\alpha }_2}\)  with \({\alpha }_2\in ( 1/2,{5}/{4})\) ,
by (),
\(\Vert  \mathcal {R}^B(-\Delta )^{\alpha _2} d_{q+1} \Vert _{L_t^1L^p_x} \lesssim & \Vert  \mathcal {R}^B(-\Delta )^{\alpha _2} d_{q+1}^{(p)} \Vert _{L_t^1L^p_x}+\Vert  \mathcal {R}^B(-\Delta )^{\alpha _2} d_{q+1}^{(c)} \Vert _{L_t^1L^p_x}\\& +\Vert  \mathcal {R}^B(-\Delta )^{\alpha _2} d_{q+1}^{(t)} \Vert _{L_t^1L^p_x}+\Vert  \mathcal {R}^B(-\Delta )^{\alpha _2} d_{q+1}^{(o)}\Vert _{L_t^1L^p_x}.\) 

In order to control the R.H.S. of (REF ),
using the interpolation inequality (cf. [43]}) and (REF ),
\(\Vert  \mathcal {R}^B(-\Delta )^{\alpha _2} d_{q+1}^{(p)} \Vert _{L_t^1L^p_x}& \lesssim \Vert  |{\nabla }|^{2{\alpha }_1-1} d_{q+1}^{(p)} \Vert _{L_t^1L^p_x}\\& \lesssim \Vert d_{q+1}^{(p)}\Vert _{L_t^1L^p_x} ^{\frac{3-2{\alpha }_2}{2}} \Vert d_{q+1}^{(p)}\Vert _{L_t^1W^{2,p}_x} ^{\frac{2{\alpha }_2-1}{2}}\\& \lesssim \ell ^{-1}\lambda ^{2\alpha _2-1}r_{\perp }^{\frac{1}{p}-\frac{1}{2}}r_{\parallel }^{\frac{1}{p}-\frac{1}{2}}\tau ^{-\frac{1}{2}}.\) 

Similarly, by Lemma REF ,
\(&\Vert  \mathcal {R}^B(-\Delta )^{\alpha _2} d_{q+1}^{(c)} \Vert _{L_t^1L^p_x}\lesssim \ell ^{-1}\lambda ^{2\alpha _2-1}r_{\perp }^{\frac{1}{p}+\frac{1}{2}}r_{\parallel }^{\frac{1}{p}-\frac{3}{2}}\tau ^{-\frac{1}{2}},\\&\Vert  \mathcal {R}^B(-\Delta )^{\alpha _2} d_{q+1}^{(t)} \Vert _{L_t^1L^p_x}\lesssim \ell ^{-2}\lambda ^{2\alpha _2-1}\mu ^{-1}r_{\perp }^{\frac{1}{p}-1}r_{\parallel }^{\frac{1}{p}-1},\\&\Vert  \mathcal {R}^B(-\Delta )^{\alpha _2} d_{q+1}^{(o)}\Vert _{L_t^1L^p_x} \lesssim \ell ^{-14}\sigma ^{-1}.\) 

Hence, we conclude from (REF )-()
and the fact that \(2\alpha _2-1 \le {3}/{2}-10\varepsilon \)  that
\( \Vert  \mathcal {R}^B(-\Delta )^{\alpha _2} d_{q+1} \Vert _{L_t^1L^p_x}& \lesssim \ell ^{-1}\lambda ^{2\alpha _2-1}r_{\perp }^{\frac{1}{p}-\frac{1}{2}}r_{\parallel }^{\frac{1}{p}-\frac{1}{2}}\tau ^{-\frac{1}{2}}+\ell ^{-1}\lambda ^{2\alpha _2-1}r_{\perp }^{\frac{1}{p}+\frac{1}{2}}r_{\parallel }^{\frac{1}{p}-\frac{3}{2}}\tau ^{-\frac{1}{2}} \\&\quad +\ell ^{-2}\lambda ^{2\alpha _2-1}\mu ^{-1}r_{\perp }^{\frac{1}{p}-1}r_{\parallel }^{\frac{1}{p}-1}+\ell ^{-14}\sigma ^{-1}\\& \lesssim \ell ^{-1}\lambda ^{-{\varepsilon }} .\) 
Regarding the nonlinear terms in (REF ),
estimating as in (REF ) we have
\( &\Vert  \mathcal {R}^B\mathbb {P}_H{\mathrm {div}}\left(d_{q + 1} \otimes u_{\ell } - u_{\ell } \otimes d_{q+1}+ B_{\ell }\otimes w_{q+1} -w_{q +1} \otimes B_{\ell }\right) \Vert _{L_t^1L^p_x} \nonumber \\ \lesssim \,&\Vert d_{q + 1} \otimes u_{\ell } - u_{\ell } \otimes d_{q+1}+ B_{\ell }\otimes w_{q+1} -w_{q +1} \otimes B_{\ell } \Vert _{L_t^1L^p_x} \nonumber \\\lesssim \,& \Vert u_{\ell }\Vert _{C_{t,x}} \Vert d_{q+1}\Vert _{L_t^1L^p_x} +\Vert B_{\ell }\Vert _{C_{t,x}} \Vert w_{q+1}\Vert _{L_t^1L^p_x} \nonumber \\\lesssim \, &\lambda ^7_q (\ell ^{-1} r_{\perp }^{\frac{1}{p}-\frac{1}{2}}r_{\parallel }^{\frac{1}{p}-\frac{1}{2}} \tau ^{-\frac{1}{2}}+\ell ^{-6}\sigma ^{-1} + \ell ^{-10} \sigma ^{-1} )\lesssim \ell ^{-10}\lambda ^{-2{\varepsilon }}.\) 
Therefore, combining (REF ), (REF ),
(REF ), (REF ) together
and using (REF )
and the fact that \(2\alpha _2-1\le {3}/{2}- 10 \varepsilon \)  we arrive at
\( \Vert \mathring{R}_{lin}^B \Vert _{L_t^1L^p_x}& \lesssim \ell ^{-8}\lambda ^{-2{\varepsilon }} +\ell ^{-1}\lambda ^{-{\varepsilon }}+\ell ^{-10}\lambda ^{-2{\varepsilon }}\lesssim \ell ^{-1}\lambda ^{-{\varepsilon }}.\) 

Oscillation error.
In order to treat the delicate magnetic oscillations,
we decompose
\(\mathring{R}_{osc}^B = \mathring{R}_{osc.1}^B + \mathring{R}_{osc.2}^B+ \mathring{R}_{osc.3}^B,\) 

where \(\mathring{R}_{osc.1}^B\)  contains the low-high spatial oscillations
\(\mathring{R}_{osc.1}^B&:= \sum _{k \in \Lambda _B}\mathcal {R}^B \mathbb {P}_{H}\mathbb {P}_{\ne 0}\left(g_{(\tau )}^2 \mathbb {P}_{\ne 0}(D_{(k)}\otimes W_{(k)}-W_{(k)}\otimes D_{(k)})\nabla (a_{(k)}^2) \right),\) 

\(\mathring{R}_{osc.2}^B\)  contains the high temporal oscillation
\(\mathring{R}_{osc.2}^B&:= - \mu ^{-1} \sum _{k \in \Lambda _B}\mathcal {R}^B\mathbb {P}_{H}\mathbb {P}_{\ne 0}\left(\partial _t (a_{(k)}^2g_{(\tau )}^2) \psi _{(k_1)}^2\phi _{(k)}^2k_2\right),\) 

and \(\mathring{R}_{osc.3}^B\)  is of low frequency
\(\mathring{R}_{osc.3}^B &:= -\sigma ^{-1}\sum _{k\in \Lambda _B}\mathcal {R}^B\mathbb {P}_{H}\mathbb {P}_{\ne 0}\left(h_{(\tau )}{}_{3}D_{(k)}\otimes W_{(k)}-W_{(k)}\otimes D_{(k)}{\rm d}x\partial _t\nabla (a_{(k)}^{2})\right).\) 
In order to estimate \(\mathring{R}_{osc.1}^B \) ,
the key fact is that,
the velocity and magnetic flows are of high oscillations
\(\mathbb {P}_{\ne 0} (D_{(k)} \otimes D_{(k)} - W_{(k)} \otimes W_{(k)})= \mathbb {P}_{\ge \frac{1}{2} \lambda r_{\perp }} (D_{(k)} \otimes D_{(k)} - W_{(k)} \otimes W_{(k)}),\) 

while the amplitude function \(a_{(k)}\)  is slowly varying.
Hence, intuitively,
the frequency of \(\mathring{R}^B_{osc.1}\)  concentrates at the high mode,
and thus the inverse-divergence operator \(\mathcal {R}^B\)  permits to gain
a small factor \((\lambda r_{\perp })^{-1}\) .
This is the content of Lemma REF  below.
Lemma 5.3 ([44]}, Lemma 6; see also [11]}, Lemma B.1)  
Let \(a \in C^{2}\left(\mathbb {T}^{3}\right)\) . For all \(1<p<+\infty \)  we have
\(\left\Vert |\nabla |^{-1} \mathbb {P}_{\ne 0}\left(a \mathbb {P}_{\ge k} f\right)\right\Vert _{L^{p}\left(\mathbb {T}^{3}\right)} \lesssim k^{-1}\left\Vert \nabla ^{2} a\right\Vert _{L^{\infty }\left(\mathbb {T}^{3}\right)}\Vert f\Vert _{L^{p}\left(\mathbb {T}^{3}\right)},\) 
holds for any smooth function \(f \in L^{p}\left(\mathbb {T}^{3}\right)\) .

We apply Lemma REF 
with \(a = \nabla (a_{(k)}^2)\)  and \(f = \psi _{(k_1)}^2\phi _{(k)}^2\) 
to get
\( \Vert \mathring{R}_{osc.1}^B \Vert _{L^1_tL^p_x}&\lesssim \sum _{ k \in \Lambda _B}\Vert g_{(\tau )}\Vert _{L^2_t}^2\Vert |\nabla |^{-1} \mathbb {P}_{\ne 0}\left(\mathbb {P}_{\ge (\lambda r_{\perp }/2)}(D_{(k)}\otimes W_{(k)}-W_{(k)}\otimes D_{(k)} )\nabla (a_{(k)}^2)\right)\Vert _{C_tL^p_x} \nonumber \\& \lesssim \sum _{ k \in \Lambda _B} \lambda ^{-1} r_{\perp }^{-1} \Vert  {\nabla }^3(a^2_{(k)})\Vert _{C_{t,x}}\Vert \psi _{(k_1)}^2\phi _{(k)}^2\Vert _{C_tL^p_x} \nonumber \\& \lesssim \sum _{ k \in \Lambda _B} \ell ^{-14} \lambda ^{-1} r_{\perp }^{-1} \Vert  \psi ^2_{(k_1)}\Vert _{C_tL^{p}_x} \Vert \phi ^2_{(k)} \Vert _{C_tL^{p}_x} \nonumber \\& \lesssim \ell ^{-14} \lambda ^{-1} r_{\perp }^{\frac{1}{p}-2}r_{\parallel }^{\frac{1}{p}-1},\) 

where we also used Lemmas REF  and REF  in the third step.
Regarding \(\mathring{R}_{osc.2}^B \) ,
as mentioned in Remark REF ,
the high temporal oscillations arising from \(g_{(\tau )}\) 
can be balanced by the large parameter \(\mu \) ,
namely, we have
\( \Vert \mathring{R}_{osc.2}^B \Vert _{L^1_tL_x^p}&\lesssim {\mu }^{-1} \sum _{k\in \Lambda _B}\Vert |\nabla |^{-1} \mathbb {P}_{H} \mathbb {P}_{\ne 0}\left(\partial _t (a_{(k)}^2g_{(\tau )}^2)\psi _{(k_1)}^2\phi _{(k)}^2k_2\right)\Vert _{L^1_tL_x^p} \nonumber \\&\lesssim {\mu }^{-1} \sum _{k\in \Lambda _B}\left( \Vert \partial _t (a_{(k)}^2) \Vert _{C_{t,x}}\Vert g_{(\tau )}^2 \Vert _{L^1_t}+ \Vert a_{(k)} \Vert _{C_{t,x}}^2\Vert \partial _t(g_{(\tau )}^2)\Vert _{ L_t^1 } \right)\Vert \psi _{(k_1)}\Vert _{C_tL^{2p}_x}^2\Vert \phi _{(k)}\Vert _{L^{2p}_x}^2 \nonumber \\&\lesssim (\ell ^{-6}+\ell ^{-2}\tau \sigma )\mu ^{-1}r_{\perp }^{\frac{1}{p}-1}r_{\parallel }^{\frac{1}{p}-1}\lesssim \ell ^{-6}\tau \sigma \mu ^{-1}r_{\perp }^{\frac{1}{p}-1}r_{\parallel }^{\frac{1}{p}-1}.\) 
Finally, the low frequency part \(\mathring{R}_{osc.3}^B \) 
can be estimated easily by (REF ) and (),
\( \Vert \mathring{R}_{osc.3}^B \Vert _{L^1_tL^p_x}&\lesssim \sigma ^{-1} \sum _{k\in \Lambda _B}\left\Vert h_{(\tau )}(k_2\otimes k_1-k_1\otimes k_2)\partial _t\nabla (a_{(k)}^{2})\right\Vert _{L^1_tL^p_x} \nonumber \\&\lesssim \sigma ^{-1} \sum _{k\in \Lambda _B} \Vert h_{(\tau )}\Vert _{C_t}\left( \Vert a_{(k)} \Vert _{C_{t,x}} \Vert a_{(k)} \Vert _{C_{t,x}^2} +\Vert a_{(k)} \Vert _{C_{t,x}^1}^2\right)\nonumber \\&\lesssim \ell ^{-9} \sigma ^{-1}.\) 
Therefore, putting estimates (REF )-(REF ) altogether
and using (REF ), (REF ) we obtain
\(\Vert \mathring{R}_{osc}^B\Vert _{L_t^1L^p_x}&\lesssim \ell ^{-14} \lambda ^{-1} r_{\perp }^{\frac{1}{p}-2}r_{\parallel }^{\frac{1}{p}-1}+\ell ^{-6}\tau \sigma \mu ^{-1}r_{\perp }^{\frac{1}{p}-1}r_{\parallel }^{\frac{1}{p}-1}+\ell ^{-9} \sigma ^{-1} \\&\lesssim \ell ^{-14}\lambda ^{-\varepsilon }+\ell ^{-6}\lambda ^{-\frac{1}{2}+3\varepsilon }+\ell ^{-9} \lambda ^{-2\varepsilon } \\& \lesssim \ell ^{-14} \lambda ^{-\varepsilon }.\) 

Corrector error.
We take \(p_1,p_2\in (1,{\infty })\)  such that
\(1/p_1=1-\eta \)  with \(\eta \)  small enough such that
\(\eta \le {\varepsilon }/(2(2-8{\varepsilon }))\) ,
and \(1/{p_1}={1}/{p_2}+1/2\) .
Then,
using Hölder's inequality,
applying Lemma REF  to (REF )
and using (REF ), () we get
\(\Vert \mathring{R}_{cor}^B \Vert _{L^1_{t}L^{p_1}_x}\lesssim & \Vert  d_{q+1}^{(p)} \otimes (w_{q+1}^{(c)}+ w_{q+1}^{(t)}+w_{q+1}^{(o)}) -(w_{q+1}^{(c)}+w_{q+1}^{(t)}+w_{q+1}^{(o)}) \otimes d_{q+1}\\&\quad + (d_{q+1}^{(c)}+d_{q+1}^{(t)}+d_{q+1}^{(o)})\otimes w_{q+1} -w_{q+1}^{(p)} \otimes (d_{q+1}^{(c)}+d_{q+1}^{(t)}+d_{q+1}^{(o)}) \Vert _{L^1_{t}L^{p_1}_x} \\\lesssim & \Vert w_{q+1}^{(c)}+w_{q+1}^{(t)}+w_{q+1}^{(o)}\Vert _{L^2_{t}L^{p_2}_x} (\Vert d^{(p)}_{q+1} \Vert _{L^2_{t,x}} + \Vert d_{q+1} \Vert _{L^2_{t,x}}) \\& + (\Vert  w_{q+1}^{(p)}\Vert _{L^2_{t,x}} + \Vert  w_{q+1}\Vert _{L^2_{t,x}}) \Vert  d_{q+1}^{(c)}+ d_{q+1}^{(t)}+d_{q+1}^{(o)}\Vert _{L^2_{t}L^{p_2}_x}\\\lesssim & \left( \ell ^{-1}r_{\perp }^{\frac{1}{p_2}+\frac{1}{2}}r_{\parallel }^{\frac{1}{p_2}-\frac{3}{2}}+ \ell ^{-2}\mu ^{-1} r_{\perp }^{\frac{1}{p_2}-1}r_{\parallel }^{\frac{1}{p_2}-1} \tau ^{\frac{1}{2}} +\ell ^{-10}\sigma ^{-1}\right) \\& \times \left(\delta _{q+1}^\frac{1}{2} + \ell ^{-1} r_{\perp }r_{\parallel }^{-1} + \ell ^{-2} \mu ^{-1} r_{\perp }^{-\frac{1}{2}} r_{\parallel }^{-\frac{1}{2}} \tau ^{\frac{1}{2}}+ \ell ^{-10} \sigma ^{-1}\right) \\\lesssim & \ell ^{-20}\delta _{q+1}^\frac{1}{2} (\lambda ^{-4{\varepsilon }+2\eta -8\eta {\varepsilon }}+\lambda ^{-{\varepsilon }+2\eta -8\eta {\varepsilon }}+ \lambda ^{-2{\varepsilon }} ) \lesssim \ell ^{-20} \lambda ^{-\frac{{\varepsilon }}{2}}, \) 

where the last step is due to the fact that
\(-{\varepsilon }/2 + 2\eta - 8\eta {\varepsilon }\le 0\) .

Commutator error.
This error can be bounded easily by using ():
\( \Vert \mathring{R}^B_{com}\Vert _{L^1_tL^p_x}\lesssim & \Vert B_{\ell } \otimes u_{\ell }-u_{\ell }\otimes B_{\ell }-\left({u_q}-{u_q}\otimes ) *_{x} \phi _{\ell } *_{t} \varphi _{\ell } \Vert _{L^1_tL^p_x} \\\lesssim & \ell \lambda _q^{14}.\right.\) 

Estimates of Reynolds stress
This subsection treats the Reynolds stress \(\mathring{R}_{q+1}^u\)  given by (REF ).
Linear error.
By the \(L^p\) -boundness of \(\mathcal {R}^u\) ,
() and Lemmas REF , REF -REF ,
\(&\Vert  \mathcal {R}^u\partial _t( w_{q+1}^{(p)}+w_{q+1}^{(c)})\Vert _{L_t^1L_x^p} \nonumber \\\lesssim & \sum _{k \in \Lambda _u\cup \Lambda _B}\Vert  \mathcal {R}^u {\mathrm {curl}}{\mathrm {curl}}\partial _t(g_{(\tau )}a_{(k)} W^c_{(k)}) \Vert _{L_t^1L_x^p}\nonumber \\\lesssim & \sum _{k \in \Lambda _u\cup \Lambda _B}\Vert  {\mathrm {curl}}\partial _t (g_{(\tau )}a_{(k)} W^c_k) \Vert _{L_t^1L_x^p} \nonumber \\\lesssim & \sum _{k \in \Lambda _u\cup \Lambda _B}\Big (\Vert  g_{(\tau )}\Vert _{L^1_t}(\Vert  a_{(k)} \Vert _{C_{t,x}^2}\Vert  W^c_k \Vert _{C_tW_x^{1,p}}+\Vert  a_{(k)} \Vert _{C_{t,x}^1}\Vert  \partial _t W^c_k \Vert _{C_tW^{1,p}_x}) \nonumber \\&\qquad \qquad \qquad +\Vert  \partial _tg_{(\tau )}\Vert _{L_t^1}\Vert  a_{(k)} \Vert _{C_{t,x}^1}\Vert  W^c_k \Vert _{C_tW_x^{1,p}}\Big )\nonumber \\\lesssim &\tau ^{-\frac{1}{2}}(\ell ^{-16}r_{\perp }^{\frac{1}{p}-\frac{1}{2}}r_{\parallel }^{\frac{1}{p}-\frac{1}{2}}\lambda ^{-1}+\ell ^{-8}r_{\perp }^{\frac{1}{p}+\frac{1}{2}}r_{\parallel }^{\frac{1}{p}-\frac{3}{2}}\mu )+ \sigma \tau ^{\frac{1}{2}}\ell ^{-8}r_{\perp }^{\frac{1}{p}-\frac{1}{2}}r_{\parallel }^{\frac{1}{p}-\frac{1}{2}}\lambda ^{-1} \\\lesssim & \ell ^{-16}\lambda ^{-2{\varepsilon }}.\) 
Regarding the viscosity term \((-\Delta )^{{\alpha }_1}\)  in (REF ),
arguing as in the proof of (REF ) and (REF ),
the spatial and temporal intermittencies yield that
\(\begin{aligned}\Vert  \mathcal {R}^u(-\Delta )^{\alpha _1} w_{q+1} \Vert _{L_t^1L^p_x}\lesssim \ell ^{-1}\lambda ^{-{\varepsilon }}.\end{aligned}\) 
Moreover, similarly to (REF ),
\(\Vert \mathcal {R}^u \mathbb {P}_H {\mathrm {div}}\left(u_{\ell } \mathring{\otimes } w_{q+1} + w_{q + 1} \mathring{\otimes } u_{\ell }- d_{q +1} \mathring{\otimes } B_{\ell } - B_{\ell } \mathring{\otimes } d_{q+1} \right) \Vert _{L_t^1L^p_x}\lesssim \ell ^{-10} \lambda ^{-2{\varepsilon }} .\) 
Thus, combining the above estimates together we arrive at
\( \Vert  \mathring{R}_{lin}^u \Vert _{L_t^1L^p_x}&\lesssim \ell ^{-16}\lambda ^{-2{\varepsilon }}+\ell ^{-1}\lambda ^{-{\varepsilon }}+ \ell ^{-10} \lambda ^{-2{\varepsilon }}\lesssim \ell ^{-1}\lambda ^{-\varepsilon }.\) 

Oscillation error.
For the velocity oscillation \(\mathring{R}_{osc}^u \) ,
using (REF ) we decompose
\(\mathring{R}_{osc}^u &= \mathring{R}_{osc.1}^u + \mathring{R}_{osc.2}^u+ \mathring{R}_{osc.3}^u ,\) 

where \(\mathring{R}_{osc.1}^u \)  is the low-high spatial frequency part
\(\mathring{R}_{osc.1}^u &:= \sum _{k \in \Lambda _u} \mathcal {R}^u\mathbb {P}_{\ne 0}\left(g_{(\tau )}^2 \mathbb {P}_{\ne 0}(W_{(k)}\otimes W_{(k)})\nabla (a_{(k)}^2) \right) \nonumber \\&\quad + \sum _{k \in \Lambda _B} \mathcal {R}^u \mathbb {P}_{\ne 0}\left( g_{(\tau )}^2\mathbb {P}_{\ne 0}(W_{(k)}\otimes W_{(k)}-D_{(k)}\otimes D_{(k)})\nabla (a_{(k)}^2)\right),\) 

\(\mathring{R}_{osc.2}^u \)  contains the high temporal oscillations
\(\mathring{R}_{osc.2}^u&:= - {\mu }^{-1} \sum _{k \in \Lambda _u \cup \Lambda _B}\mathcal {R}^u \mathbb {P}_{\ne 0}\left(\partial _t (a_{(k)}^2g_{(\tau )}^2)\psi _{(k_1)}^2\phi _{(k)}^2k_1\right),\) 

and \(\mathring{R}_{osc.3}^u\)  is of low frequency
\(\mathring{R}_{osc.3}^u:=&- \sigma ^{-1}\sum _{k\in \Lambda _u}\mathcal {R}^u \mathbb {P}_{\ne 0}\left(h_{(\tau )}{}_{3}W_{(k)}\otimes W_{(k)}{\rm d}x\partial _t\nabla (a_{(k)}^{2})\right) \\&-\sigma ^{-1}\sum _{k\in \Lambda _B}\mathcal {R}^u \mathbb {P}_{\ne 0}\left(h_{(\tau )}{}_{3}W_{(k)}\otimes W_{(k)}-D_{(k)}\otimes D_{(k)}{\rm d}x\partial _t\nabla (a_{(k)}^{2})\right). \) 
Let us treat \(\mathring{R}_{osc.1}^u\) , \(\mathring{R}_{osc.2}^u\)  and \(\mathring{R}_{osc.3}^u \)  separately.
By Lemmas REF , REF , REF , REF  and REF ,
\( \Vert \mathring{R}_{osc.1}^u\Vert _{L^1_tL_x^p}& \lesssim \sum _{k \in \Lambda _u }\left\Vert g_{(\tau )}^2\right\Vert _{L^1_t}\left\Vert |{\nabla }|^{-1}\mathbb {P}_{\ne 0}\left( \mathbb {P}_{\ne 0}(W_{(k)}\otimes W_{(k)})\nabla (a_{(k)}^2) \right)\right\Vert _{L^1_tL_x^p} \nonumber \\&\quad + \sum _{k \in \Lambda _B}\left\Vert g_{(\tau )}^2\right\Vert _{L^1_t} \left\Vert |{\nabla }|^{-1}\mathbb {P}_{\ne 0}\left( \mathbb {P}_{\ne 0}(W_{(k)}\otimes W_{(k)}-D_{(k)}\otimes D_{(k)})\nabla (a_{(k)}^2)\right)\right\Vert _{L^1_tL_x^p} \\& \lesssim \sum _{k \in \Lambda _u \cup \Lambda _B } (\lambda r_{\perp })^{-1} \Vert {\nabla }^3 (a^2_{(k)})\Vert _{C_{t,x}}\Vert \psi ^2_{(k_1)} \phi _{(k)}^2 \Vert _{C_tL^p} \\& \lesssim \ell ^{-26}\lambda ^{-1} r_{\perp }^{\frac{1}{p}-2}r_{\parallel }^{\frac{1}{p}-1} .\) 
Regarding the temporal oscillation term \(\mathring{R}_{osc.2}^u\) ,
we estimate
\( &\Vert \mathring{R}_{osc.2}^u \Vert _{L^1_tL_x^p} \\\lesssim & {\mu }^{-1}\sum _{k \in \Lambda _u \cup \Lambda _B}( \Vert \partial _t (a_{(k)}^2) \Vert _{C_{t,x}}\Vert g_{(\tau )}^2 \Vert _{L^1_t}+ \Vert a_{(k)} \Vert _{C_{t,x}}^2\Vert \partial _t(g_{(\tau )}^2)\Vert _{ L_t^1 } )\Vert \psi _{(k_1)}\Vert _{C_tL^{2p}_x}^2\Vert \phi _{(k)}\Vert _{L^{2p}_x}^2 \nonumber \\\lesssim & (\ell ^{-10}+\ell ^{-2}\tau \sigma )\mu ^{-1}r_{\perp }^{\frac{1}{p}-1}r_{\parallel }^{\frac{1}{p}-1} \nonumber \\\lesssim & \ell ^{-10}\tau \sigma \mu ^{-1}r_{\perp }^{\frac{1}{p}-1}r_{\parallel }^{\frac{1}{p}-1}.\) 
Concerning the low frequency part \(\mathring{R}_{osc.3}^u \) ,
it holds that
\( \Vert \mathring{R}_{osc.3}^u\Vert _{L^1_tL^p_x}&\lesssim \sigma ^{-1}\sum _{k\in \Lambda _u }\left\Vert |{\nabla }|^{-1}\mathbb {P}_{\ne 0}\left(h_{(\tau )}{}_{3}W_{(k)}\otimes W_{(k)}{\rm d}x\partial _t\nabla (a_{(k)}^{2})\right)\right\Vert _{L^1_tL^p_x} \nonumber \\&\quad + \sigma ^{-1}\sum _{k\in \Lambda _B} \left\Vert  |{\nabla }|^{-1}\mathbb {P}_{\ne 0}\left(h_{(\tau )}{}_{3}W_{(k)}\otimes W_{(k)}-D_{(k)}\otimes D_{(k)}{\rm d}x\partial _t\nabla (a_{(k)}^{2})\right)\right\Vert _{L^1_tL^p_x} \nonumber \\&\lesssim \sigma ^{-1} \sum _{k\in \Lambda _u\cup \Lambda _B}\Vert h_{(\tau )}\Vert _{C_t} \left( \Vert a_{(k)} \Vert _{C_{t,x}} \Vert a_{(k)} \Vert _{C_{t,x}^2} +\Vert a_{(k)} \Vert _{C_{t,x}^1}^2\right)\nonumber \\&\lesssim \ell ^{-17} \sigma ^{-1}.\) 
Therefore, putting estimates (REF )-(REF ) altogether
we arrive at
\( \Vert \mathring{R}_{osc}^u \Vert _{L_t^1L^p_x}&\lesssim \ell ^{-26}\lambda ^{-1} r_{\perp }^{\frac{1}{p}-2}r_{\parallel }^{\frac{1}{p}-1}+\ell ^{-10}\tau \sigma \mu ^{-1}r_{\perp }^{\frac{1}{p}-1}r_{\parallel }^{\frac{1}{p}-1}+ \ell ^{-17} \sigma ^{-1} \\& \lesssim \ell ^{-26}\lambda ^{-\varepsilon }.\) 

Corrector error.
Take \(p_1, p_2\in (1,{\infty })\)  as in the proof of (REF ).
Applying Lemma REF  and using (REF ) and (),
we have, similarly to (REF ),
\( \Vert \mathring{R}_{cor}^u \Vert _{L^1_{t}L^{p_1}_x} &\lesssim \Vert w_{q+1}^{(c)}+w_{q+1}^{(t)}+ w_{q+1}^{(o)}\Vert _{L^2_{t}L^{p_2}_x}(\Vert w^{(p)}_{q+1} \Vert _{L^2_{t,x}} + \Vert w_{q+1} \Vert _{L^2_{t,x}})\\&\quad + (\Vert d^{(p)}_{q+1} \Vert _{L^2_{t,x}} + \Vert d_{q+1} \Vert _{L^2_{t,x}}) \Vert  d_{q+1}^{(c)}+ d_{q+1}^{(t)}+d_{q+1}^{(o)} \Vert _{L^2_{t}L^{p_2}_x} \\& \lesssim \ell ^{-20} \lambda ^{-\frac{{\varepsilon }}{2}}.\) 

Commutator error.
Using () one has
\( \Vert \mathring{R}^u_{com}\Vert _{L^1_tL^p_x}\lesssim & \Vert u_{\ell }\mathring{\otimes } u_{\ell }-B_{\ell } \mathring{\otimes }B_{\ell }-\left({u_q}\mathring{\otimes }{u_q}-{\otimes }) *_{x} \phi _{\ell } *_{t} \varphi _{\ell } \Vert _{L^1_tL^p_x} \\\lesssim & \ell \lambda _q^{14}.\right.\) 

Proof of main iteration in Theorem  REF
The inductive estimates (REF ), (REF ) and (REF )
of the velocity and magnetic fields
have been verified in Subsection REF .
Hence, it remains to verify the estimates () and ()
for the stresses and to verify the inductive inclusions (REF ) and () for the temporal supports.
(i) Verification of the inductive estimates () and ().
Regarding the inductive estimates in (),
by the identity (REF ),
Sobolev's embedding \(W^{1,6}_x \hookrightarrow L^{{\infty }}_x\) ,
\(\Vert \mathring{R}^u_{q+1}\Vert _{C_tC^1_x}&\lesssim \Vert \mathcal {R}^u \mathbb {P}_H ({\mathrm {div}}\mathring{R}^u_{q+1})\Vert _{C_tW^{2,6}_x}\\&\lesssim \Vert \partial _t u_{q+1}+\nu _1 (-\Delta )^{\alpha _1} u_{q+1}+{\mathrm {div}}(u_{q+1}\otimes u_{q+1}-B_{q+1}\otimes B_{q+1}) \Vert _{C_tW^{1,6}_x}.\) 

Then, using the interpolation inequality (cf. [43]}),
(), () and ()
we get
\(\Vert \mathring{R}^u_{q+1}\Vert _{C_tC^1_x}\lesssim & \Vert u_{q+1}\Vert _{C^2_{t,x}}+ \Vert  u_{q+1}\Vert _{C_tL^6_x}^{\frac{3-2{\alpha }_1}{4}} \Vert  u_{q+1}\Vert _{C_t W^{4,6}_x}^{\frac{2{\alpha }_1+1}{4}} \\& + \sum \limits _{0\le N_1+N_2\le 2}\left(\Vert u_{q+1}\Vert _{C^{N_1}_{t,x}}\Vert u_{q+1}\Vert _{C^{N_2}_{t,x}}+ \Vert B_{q+1}\Vert _{C^{N_1}_{t,x}}\Vert B_{q+1}\Vert _{C^{N_2}_{t,x}} \right) \\\lesssim & \lambda _{q+1}^{14}. \) 

Similarly,
\(\Vert \partial _t \mathring{R}^u_{q+1}\Vert _{C_{t,x}}&\lesssim \Vert \partial _t^2 u_{q+1}+\nu _1 \partial _t(-\Delta )^{\alpha _1} u_{q+1}+{\mathrm {div}}\partial _t(u_{q+1}\otimes u_{q+1}-B_{q+1}\otimes B_{q+1}) \Vert _{C_{t}L^6_x} \\&\lesssim \Vert u_{q+1}\Vert _{C^2_{t,x}}+ \Vert  u_{q+1}\Vert _{C^4_{t,x}}+ \Vert  u_{q+1} \otimes u_{q+1}\Vert _{C^2_{t,x}}+ \Vert  B_{q+1} \otimes B_{q+1}\Vert _{C^2_{t,x}} \\&\lesssim \lambda _{q+1}^{14}.\) 
For the \(C_{t, x}^1\)  estimate of \(\mathring{R}_{q+1}^B\) ,
using the identity (REF )
Sobolev's embedding \(W^{1,6}_x\hookrightarrow L^{\infty }_x\) ,
the interpolation inequality
and then using () and
() we obtain
\(\Vert \mathring{R}^B_{q+1}\Vert _{C_tC^1_x}& \lesssim \Vert \mathcal {R}^B\mathbb {P}_H ({\mathrm {div}}\mathring{R}^B_{q+1})\Vert _{C_tW^{2,6}_x}\\&\lesssim \Vert \partial _t B_{q+1}+\nu _2 (-\Delta )^{\alpha _2} B_{q+1}+{\mathrm {div}}(B_{q+1}\otimes u_{q+1}-u_{q+1}\otimes B_{q+1}) \Vert _{C_tW^{1,6}_x}\\&\lesssim \Vert \partial _t B_{q+1}\Vert _{{C_tC^1_x}}+ \Vert B_{q+1}\Vert _{C_tL^6_x}^{\frac{3-2{\alpha }_2}{4}} \Vert B_{q+1}\Vert _{C_tW^{4,6}_x}^{\frac{2{\alpha }_2+1}{4}} \\&\quad +\Vert B_{q+1}\otimes u_{q+1}- u_{q+1}\otimes B_{q+1}\Vert _{C_tC^{2}_x}\\&\lesssim \lambda _{q+1}^{14}, \) 

and, similarly,
\(\Vert \partial _t \mathring{R}^B_{q+1}\Vert _{C_{t,x}}&\lesssim \Vert \partial _t^2 B_{q+1}+\nu _2 \partial _t(-\Delta )^{\alpha _2} B_{q+1}+{\mathrm {div}}\partial _t(B_{q+1}\otimes u_{q+1}-u_{q+1}\otimes B_{q+1}) \Vert _{C_{t}L^6_x}\\&\lesssim \lambda _{q+1}^{14}. \) 
Thus,
the estimates in () are verified.
Moreover, combining (REF ),
(REF ), (REF )
and (REF )
we obtain
\( \Vert \mathring{R}_{q+1}^u \Vert _{L^1_{t,x}}&\le \Vert  \mathring{R}_{lin}^u \Vert _{L_t^1L^p_x} + \Vert  \mathring{R}_{osc}^u\Vert _{L_t^1L^p_x} + \Vert \mathring{R}_{cor}^u \Vert _{L^1_{t}L^{p_1}_x}+ \Vert \mathring{R}_{com}^u \Vert _{L_{t}^1L^p_x} \nonumber \\&\lesssim \ell ^{-1}\lambda _{q+1}^{-\varepsilon }+\ell ^{-26}\lambda _{q+1}^{-\varepsilon }+ \ell ^{-20}\lambda _{q+1}^{-\frac{{\varepsilon }}{2}} +\ell \lambda _q^{14} \nonumber \\& \le \delta _{q+2}.\) 

Combining (REF ),
(REF ),
(REF )
and (REF ) altogether we also get
\( \Vert \mathring{R}_{q+1}^B \Vert _{L^1_{t,x}}&\le \Vert  \mathring{R}_{lin}^B \Vert _{L_t^1L^p_x} + \Vert  \mathring{R}_{osc}^B\Vert _{L_t^1L^p_x}+ \Vert \mathring{R}_{cor}^B \Vert _{L^1_{t}L^{p_1}_x} + \Vert \mathring{R}_{com}^B \Vert _{L_{t}^1L^p_x} \nonumber \\&\lesssim \ell ^{-1}\lambda _{q+1}^{-\varepsilon }+\ell ^{-14}\lambda _{q+1}^{-\varepsilon } + \ell ^{-20}\lambda _{q+1}^{-\frac{{\varepsilon }}{2}} +\ell \lambda _q^{14} \nonumber \\& \le \delta _{q+2},\) 

where the last step is due to (REF ) and (REF ).
Therefore, (REF ) and (REF ) together verify the estimates in ().
(ii) Verification of the inductive inclusions (REF ) and ().
By definitions,
\(& \mathrm {supp\,}_t w_{q+1} \subseteq \bigcup _{k\in \Lambda _{u}\cup \Lambda _{B}}\mathrm {supp\,}_t a_{(k)} \subseteq N_{3\ell } (\mathrm {supp\,}_t \mathring{R}_{q}^u\cup \mathrm {supp\,}_t \mathring{R}_{q}^B ), \\& \mathrm {supp\,}_t d_{q+1} \subseteq \bigcup _{k\in \Lambda _{B}}\mathrm {supp\,}_t a_{(k)} \subseteq N_{3\ell } ( \mathrm {supp\,}_t \mathring{R}_{q}^u\cup \mathrm {supp\,}_t \mathring{R}_{q}^B ), \) 


which yield that
\(& \mathrm {supp\,}_t u_{q+1} \subseteq \mathrm {supp\,}_t u_{\ell } \cup \mathrm {supp\,}_t w_{q+1}\subseteq N_{3\ell } (\mathrm {supp\,}_t u_{q} \cup \mathrm {supp\,}_t \mathring{R}_{q}^u\cup \mathrm {supp\,}_t \mathring{R}_{q}^B ), \\& \mathrm {supp\,}_t B_{q+1} \subseteq \mathrm {supp\,}_t B_{\ell } \cup \mathrm {supp\,}_t d_{q+1} \subseteq N_{3\ell } (\mathrm {supp\,}_t B_{q} \cup \mathrm {supp\,}_t \mathring{R}_{q}^u\cup \mathrm {supp\,}_t \mathring{R}_{q}^B ). \) 


Thus,
taking into account \(3\ell \ll \delta ^\frac{1}{2} _{q+1}\) 
we verify the inductive inclusion in (REF ).
Moreover, by (REF ) and (REF ),
\(& \mathrm {supp\,}_t \mathring{R}^u_{q+1}\subseteq \bigcup \limits _{k\in \Lambda _u \cup \Lambda _B}\mathrm {supp\,}_t a_{(k)}\cup \mathrm {supp\,}_t u_l \cup \mathrm {supp\,}_t B_l , \\& \mathrm {supp\,}_t \mathring{R}^B_{q+1}\subseteq \bigcup \limits _{k\in \Lambda _u \cup \Lambda _B}\mathrm {supp\,}_t a_{(k)}\cup \mathrm {supp\,}_t u_l \cup \mathrm {supp\,}_t B_l . \) 

In view of (REF ) and (REF ),
we arrive at
\( \mathrm {supp\,}_t \mathring{R}^u_{q+1} \cup \mathrm {supp\,}_t \mathring{R}^B_{q+1}\subseteq N_{3\ell }\left( \mathrm {supp\,}_t u_{q} \cup \mathrm {supp\,}_t B_{q}\cup \mathrm {supp\,}_t \mathring{R}_{q}^u\cup \mathrm {supp\,}_t \mathring{R}_{q}^B\right).\) 

Since \(\ell \ll \delta _{q+2}^{\frac{1}{2}}\)  due to (REF ),
we then verify the inductive inclusion ().
Therefore, the proof of Theorem REF  is complete.
\(\square \)

Proof of Main Theorems 
Proof of Theorem  REF
We prove the statements \((i)\) -\((v)\)  in Theorem REF  below.
\((i)\) . In the initial step \(q=0\) ,
we take \(u_0=\tilde{u}\)  and \(B_0=\tilde{B}\) 
and define \(P_0\) , \(\mathring{R}_{0}^u\)  and \(\mathring{R}_{0}^B\)  in the relaxation equation (REF ) by
\(& P_0 = -\frac{1}{3} (|u_0|^2 - |B_0|^2),\\&\mathring{R}_0^u=\mathcal {R}^u\left(\partial _t u_0+\nu _1(-\Delta )^{\alpha _1} u_0\right) + u_0\mathring{\otimes }u_0-B_0\mathring{\otimes }B_0, \\& \mathring{R}_0^B=\mathcal {R}^B\left(\partial _t B_0+\nu _2(-\Delta )^{\alpha _2} B_0\right) + B_0\otimes u_0-u_0\otimes B_0. \) 
Below we choose \(a,\,M\)  sufficiently large and set
\(\delta _{1}:= \max \lbrace  \Vert \mathring{R}_0^u \Vert _{L^1_{t,x}}, \Vert \mathring{R}_0^B \Vert _{L^1_{t,x}} \rbrace .\) 

Then, (REF )-() are satisfied at level \(q=0\) . Thus,
by Theorem REF ,
there exists a sequence of solutions \((u_{q+1},B_{q+1},\mathring{R}_{q+1}^u,\mathring{R}_{q+1}^B)\)  to (REF )
obeying estimates (REF )-() for all \(q\ge 0\) .
Then, by the interpolation inequality
and (REF ), (REF ) and (REF ), for any \(\beta ^{\prime }\in (0,\frac{\beta }{7+\beta })\) ,
\(&\,\sum _{q \ge 0} \Vert  u_{q+1} - u_q \Vert _{H^{\beta ^{\prime }}_{t,x}} +\sum _{q \ge 0} \Vert  B_{q+1} - B_q \Vert _{H^{\beta ^{\prime }}_{t,x}}\\\le & \, \sum _{q \ge 0} \Vert  u_{q+1} - u_q \Vert _{L^2_{t,x}}^{1- \beta ^{\prime }}\Vert  u_{q+1} - u_q \Vert _{H^1_{t,x}}^{\beta ^{\prime }} +\sum _{q \ge 0} \Vert  B_{q+1} - B_q \Vert _{L^2_{t,x}}^{1- \beta ^{\prime }}\Vert  B_{q+1} - B_q \Vert _{H^1_{t,x}}^{\beta ^{\prime }}\\\lesssim &\, \sum _{q \ge 0} M^{1-\beta ^{\prime }} \delta _{q+1}^{\frac{1-\beta ^{\prime }}{2}}\lambda _{q+1}^{7 \beta ^{\prime } } \lesssim M^{1-\beta ^{\prime }} \delta _{1}^{\frac{1-\beta ^{\prime }}{2}}\lambda _{1}^{7 \beta ^{\prime } } +\sum _{ q \ge 1} M^{1-\beta ^{\prime }} \lambda _{q+1}^{-\beta (1 - \beta ^{\prime }) + 7\beta ^{\prime } } <{\infty }, \) 

where the last step is due to the fact that
\(-\beta (1 - \beta ^{\prime }) + 7\beta ^{\prime } <0\) .
Thus, \(\lbrace (u_q,B_q)\rbrace _{q\ge 0}\)  is a Cauchy sequence in \(H^{\beta ^{\prime }}_{t,x}\) 
and \(\lim _{q\rightarrow +\infty }(u_q,B_q)=(u,B)\)  for some \(u,B\in H^{\beta ^{\prime }}_{t,x}\) .
Taking into account
\(\lim _{q \rightarrow +\infty } \mathring{R}_{q}^u = \lim _{q \rightarrow +\infty } \mathring{R}_{q}^B = 0 \)  in \(L^1_{t,x}\) 
we consequently conclude that \((u,B)\)  is a weak solution to (REF ) in the sense of Definition REF .
\((ii)\) .
Concerning the regularity of \((u,B)\) , we first claim that for all \(q\ge 0\) ,
\(\Vert u_q\Vert _{C^4_{t,x}}\le \lambda _{q}^{14},\ \ \Vert B_q\Vert _{C^4_{t,x}}\le \lambda _{q}^{14}.\) 

To this end, for \(a\)  sufficiently large,
(REF ) holds at level \(q=0\) .
For \(q\ge 1\) , assuming that (REF ) is correct at level \(q\) ,
we apply (REF ), () and () to get
\(& \Vert u_{q+1}\Vert _{C^4_{t,x}}\le \Vert u_{\ell }\Vert _{C^4_{t,x}}+\Vert w_{q+1}\Vert _{C^4_{t,x}}\le \lambda _{q}^{14}+ \frac{1}{2}\lambda _{q+1}^{14}\le \lambda _{q+1}^{14}, \\& \Vert B_{q+1}\Vert _{C^4_{t,x}}\le \Vert B_{\ell }\Vert _{C^4_{t,x}}+\Vert d_{q+1}\Vert _{C^4_{t,x}}\le \lambda _{q}^{14}+\frac{1}{2}\lambda _{q+1}^{14}\le \lambda _{q+1}^{14},\) 

which yields (REF ) by inductive arguments, as claimed.
Therefore, for any \((s,p,\gamma )\in \mathcal {A}\) ,
using (REF ), (REF ), (REF )
and Lemma REF  we have
\(\sum _{q \ge 0}\Vert  u_{q+1} - u_q \Vert _{L^\gamma _tW^{s,p}_x} & \le \sum _{q \ge 0}\Vert u_{\ell }-u_q\Vert _{L^\gamma _tW^{s,p}_x}+\sum _{q \ge 0}\Vert w_{q+1}\Vert _{L^\gamma _tW^{s,p}_x}\\&\lesssim \sum _{q \ge 0}( \ell \Vert u_q\Vert _{C_{t,x}^4} + \ell ^{-1}\lambda _{q+1}^{s}r_{\perp }^{\frac{1}{p}-\frac{1}{2}}r_{\parallel }^{\frac{1}{p}-\frac{1}{2}}\tau ^{\frac{1}{2}-\frac{1}{\gamma }}+{\ell }^{-26}\sigma ^{-1})\\&\lesssim \sum _{q \ge 0} (\ell \lambda _{q}^{14}+ \ell ^{-1}\lambda _{q+1}^{s}r_{\perp }^{\frac{1}{p}-\frac{1}{2}}r_{\parallel }^{\frac{1}{p}-\frac{1}{2}}\tau ^{\frac{1}{2}-\frac{1}{\gamma }}+{\ell }^{-26}\lambda _{q+1}^{-2{\varepsilon }} )\\&\lesssim \sum _{q \ge 0} (\lambda _{q}^{-6}+ \lambda _{q+1}^{s+\frac{3}{2}-\frac{2}{p}-\frac{1}{\gamma } +{\varepsilon }(\frac{8}{p}+\frac{6}{\gamma }- 6) }+ \lambda _{q+1}^{-{\varepsilon }} ). \) 

Analogous arguments also yield that
\(\sum _{q \ge 0}\Vert B_{q+1} - B_q \Vert _{L^\gamma _tW^{s,p}_x}&\lesssim \sum _{q \ge 0}(\lambda _{q}^{-6}+ \lambda _{q+1}^{s+\frac{3}{2}-\frac{2}{p}-\frac{1}{\gamma } +{\varepsilon }\left(\frac{8}{p}+\frac{6}{\gamma }- 6\right) } + \lambda _{q+1}^{-{\varepsilon }}). \) 

Taking into account (REF ) we have
\(s+\frac{3}{2}-\frac{2}{p}-\frac{1}{\gamma } +{\varepsilon }\left(\frac{8}{p}+\frac{6}{\gamma }- 6\right)\le s+\frac{3}{2}-\frac{2}{p}-\frac{1}{\gamma } +8 {\varepsilon }<0,\) 
which yields that
\(\sum _{q \ge 0}\Vert  u_{q+1} - u_q \Vert _{L^\gamma _tW^{s,p}_x}+\sum _{q \ge 0}\Vert B_{q+1} - B_q \Vert _{L^\gamma _tW^{s,p}_x} < {\infty }. \) 

Therefore, \(\lbrace (u_q,B_q)\rbrace _{q\ge 0}\)  is a Cauchy sequence in \(L^\gamma _tW^{s,p}_x \times L^\gamma _tW^{s,p}_x\) 
for any \((s,p,\gamma )\in \mathcal {A}\) .
Using the uniqueness of weak limits we then conclude that
\(u, B \in H^{\beta ^\prime }_{t,x} \cap L^\gamma _tW^{s,p}_x\) .
Thus, the regularity statement \((ii)\)  is proved.
\((iii).\)  Concerning the temporal support,
let
\(K_q:= \mathrm {supp\,}_t u_q \cup \mathrm {supp\,}_t B_q\cup \mathrm {supp\,}_t \mathring{R}^u_q\cup \mathrm {supp\,}_t \mathring{R}^B_q, \ \ q\ge 1.\) 

By (REF ) and (),
\(& \mathrm {supp\,}_t \mathring{R}^u_0 \cup \mathrm {supp\,}_t \mathring{R}^B_0\subseteq K_0 := \mathrm {supp\,}_t u_0 \cup \mathrm {supp\,}_t B_0 .\) 

Moreover, by (REF ) and (),
\(K_{q+1} \subseteq N_{\delta _{q+2}^\frac{1}{2}} K_{q}\subseteq \cdots \subseteq N_{\sum \limits _{j=2}^{q+2}\delta _{j}^\frac{1}{2}} K_{0},\) 

which along with the fact that
\(\sum _{q\ge 0}\delta _{q+2}^{1/2}\le {\varepsilon }_*\)  yields that
\(&\mathrm {supp\,}_t u\cup \mathrm {supp\,}_t B\subseteq \bigcup _{q\ge 0} K_q\subseteq N_{{\varepsilon }_*} ( \mathrm {supp\,}_t \tilde{u} \cup \mathrm {supp\,}_t \tilde{B} ),\) 

thereby yielding the temporal support statement \((iii)\) .
\((iv).\)  Using (REF ) and (REF )
we get that for \(a\)  sufficiently large (depending on \({\varepsilon }_*\) ),
\(& \Vert  u - \tilde{u} \Vert _{L^1_tL^2_x} +\Vert  B - \tilde{B} \Vert _{L^1_tL^2_x} \\\le &\,\sum _{q \ge 0} \Vert  u_{q+1} - u_q \Vert _{L^1_tL^2_x} +\sum _{q \ge 0} \Vert  B_{q+1} - B_q \Vert _{L^1_tL^2_x} \\\le &\,2\sum _{q\ge 0} \delta _{q+2} ^{\frac{1}{2}} \le 2\sum _{q\ge 2} a^{-\beta b^q} \le 2\sum _{q\ge 2} a^{-\beta bq}=\frac{2a^{-2\beta b}}{1-a^{-\beta b}}\le {\varepsilon }_* , \) 

which yields the small deviation statement \((iv)\) .
\((v).\)  It remains to prove the small deviations of magnetic helicity.
Note that
\(\Vert \mathcal {H}_{B,B}-\mathcal {H}_{\widetilde{B},\widetilde{B} }\Vert _{L_t^1}\le \Vert B-B_0\Vert _{L^1_tL^2_{x}}\Vert A_0\Vert _{C_tL^2_x}+\Vert A-A_0\Vert _{L^2_{t,x}}\Vert B\Vert _{L^2_{t,x}},\) 

where \(A\)  and \(A_0\)  are the potential fields
corresponding to \(B\)  and \(B_0\) , respectively.
Since \(B_q\)  is divergence free, by the Biot-Savart law,
\(A_q:={\mathrm {curl}}(-\Delta )^{-1} B_q, \quad q\ge 0,\) 

which yields that
\(\Vert A_0\Vert _{C_tL^2_x} = \Vert {\mathrm {curl}}(-\Delta )^{-1} B_0\Vert _{C_tL^2_x} \lesssim 1.\) 
Moreover, by the inductive estimate (REF ),
\(\Vert B-B_0\Vert _{L^1_tL^2_{x}}\le & \sum _{q\ge 0}\Vert B_{q+1}-B_q\Vert _{L^1_tL^2_{x}} \le \sum _{q\ge 0} \delta _{q+2}^{\frac{1}{2}}\lesssim a^{-2\beta b}.\) 
Regarding the \(L^2\)  estimate of \(B\) , since \(\Vert B_0\Vert _{L^2_{t,x}} \lesssim 1\)  and
\(\Vert B-B_0\Vert _{L^2_{t,x}}\le & \sum _{q\ge 0}\Vert B_{q+1}-B_q\Vert _{L^2_{t,x}} \le \sum _{q\ge 0}M\delta _{q+1}^{\frac{1}{2}} \lesssim 1,\) 

we infer that
\(\Vert B\Vert _{L^2_{t,x}}\le \Vert B_0\Vert _{L^2_{t,x}}+ \Vert B-B_0\Vert _{L^2_{t,x}}\lesssim 1.\) 
Furthermore, we have,
via (REF ),
\(\Vert A-A_0\Vert _{L^2_{t,x}} &\le \sum _{q \ge 0} \Vert  A_{q+1} - A_q \Vert _{L^2_{t,x}} \le \sum _{q \ge 0} \Vert  {\mathrm {curl}}(-\Delta )^{-1}(B_{q+1} - B_q) \Vert _{L^2_{t,x}}\\&\le \sum _{q \ge 0} \Vert  {\mathrm {curl}}(-\Delta )^{-1}(B_{\ell } - B_q) \Vert _{L^2_{t,x}}+\sum _{q \ge 0} \Vert  {\mathrm {curl}}(-\Delta )^{-1}d_{q+1} \Vert _{L^2_{t,x}}.\) 

Since \(B_{\ell } - B_q\)  is mean free, by the Poincaré inequality,
\(\sum _{q \ge 0} \Vert  {\mathrm {curl}}(-\Delta )^{-1}(B_{\ell }-B_q)\Vert _{L^2_{t,x}}&\lesssim \sum _{q\ge 0} \Vert  B_{\ell }-B_q\Vert _{L^2_{t,x}}\lesssim \sum _{q \ge 0} \ell \Vert B_q\Vert _{C^1_{t,x}}\lesssim \sum _{q \ge 0} \lambda _q^{-13}\\&\lesssim a^{-13}+(a^{13b}-1)^{-1}\lesssim a^{-13}.\) 

Moreover, by (),
()
and Lemmas REF , REF ,
REF  and REF ,
\(\Vert  {\mathrm {curl}}(-\Delta )^{-1}d_{q+1}\Vert _{L^2_{t,x}} &\le \sum _{k\in \Lambda _B} \Vert  {\mathrm {curl}}(a_{(k)}g_{(\tau )}D_{(k)}^c)\Vert _{L^2_{t,x}}+ \Vert  d_{q+1}^{(t)}\Vert _{L^2_{t,x}}+ \Vert  d_{q+1}^{(o)}\Vert _{L^2_{t,x}} \\&\le \sum _{k\in \Lambda _B} \Vert  g_{(\tau )}\Vert _{L^2_t}\Vert  {\mathrm {curl}}(a_{(k)} D_{(k)}^c)\Vert _{C_tL^2_x}+ \Vert  d_{q+1}^{(t)}\Vert _{L^2_{t,x}}+ \Vert  d_{q+1}^{(o)}\Vert _{L^2_{t,x}} \\&\lesssim \sum _{k\in \Lambda _B} \Vert  g_{(\tau )}\Vert _{L^2_t} \Vert  a_{(k)}\Vert _{C^1_{t,x}} \Vert  D_{(k)}^c\Vert _{C_tW^{1,2}_x}+ \Vert  d_{q+1}^{(t)}\Vert _{L^2_{t,x}}+ \Vert  d_{q+1}^{(o)}\Vert _{L^2_{t,x}} \\&\lesssim \ell ^{-4}\lambda _{q+1}^{-1}+\ell ^{-2}\mu ^{-1}\tau ^{\frac{1}{2}}r_{\perp }^{-\frac{1}{2}}r_{\parallel }^{-\frac{1}{2}}+\ell ^{-6}\sigma ^{-1}\lesssim \lambda _{q+1}^{-\frac{{\varepsilon }}{2}}.\) 

Thus, we conclude from (REF )-(REF ) that
\(\Vert A-A_0\Vert _{L^2_{t,x}}\lesssim a^{-13}+ \sum _{q \ge 0}\lambda _{q+1}^{-\frac{{\varepsilon }}{2}}\lesssim a^{-13}+\sum _{q \ge 1} (a^{\frac{{\varepsilon }}{2}b})^{-q}\lesssim a^{-13}+ a^{-\frac{{\varepsilon }}{2} b}.\) 
Therefore, plugging (REF ), (REF ), (REF ) and (REF ) into (REF )
and using (REF )
we obtain that for \(a\)  large enough
\(\Vert \mathcal {H}_{B,B}(t)-\mathcal {H}_{\widetilde{B},\widetilde{B}}(t)\Vert _{L_t^1}&\lesssim (a^{-2\beta b} + a^{-13}+ a^{-\frac{{\varepsilon }}{2} b })\le {\varepsilon }_*.\) 
Consequently, the proof of Theorem REF  is complete.

Proof of Corollary  REF
For any \(m \in \mathbb {N}_+\) ,
we choose the incompressible, mean-free fields \(\widetilde{u}_m\)  and \(\widetilde{B}_m\) ,
defined by
\(&\widetilde{u}_m:= {m\Psi (t)} (\sin x_{3}, 0,0)^\top , \\&\widetilde{B}_m:= {m\Psi (t)} (\sin x_{3}, \cos x_{3}, 0)^\top ,\) 


where \(\Psi :{\mathbb {R}}\)  is any smooth cut-off function supported on the interval \([1/4,5/4]\) , such that
\(\Psi (t) = 1 \)  for \(1/2 \le t\le 1\)  and \(0\le \Psi (t)\le 1 \)  for all \(t\in .\) Then, for \({\varepsilon }_* \le 1/100\) ,
by Theorem REF ,
there exist \(u_m, B_m \in H^{\beta ^{\prime }}_{t,x} \cap L^\gamma _tW^{s,p}_x\) ,
where \(\beta ^{\prime }>0\)  and \((s,p,\gamma ) \in \mathcal {A}\) 
with \(\mathcal {A}\)  given by (REF ),
such that \((u_m, B_m)\)  solves (REF ) and
satisfies the properties \((ii)\) -\((v)\)  in Theorem REF .
Straightforward computations show that
\(\Vert \widetilde{u}_m\Vert _{L^1(\frac{1}{2}, 1;L^2_x)} = m\pi ^{\frac{3}{2}},\ \ \Vert \widetilde{B}_m\Vert _{L^1(\frac{1}{2}, 1;L^2_x)} = \frac{m}{2} (2\pi )^{\frac{3}{2}}.\) 

Then, in view of the small deviations on average, i.e.,
\(\Vert u_m- \widetilde{u}_m\Vert _{L^1(\frac{1}{2}, 1;L^2_x)} \le {\varepsilon }_*, \ \ \Vert B_m- \widetilde{B}_m\Vert _{L^1(\frac{1}{2}, 1;L^2_x)} \le {\varepsilon }_*,\) 

we infer that for any \(m\ne m^{\prime }\) ,
\(\Vert u_m\Vert _{L^1(\frac{1}{2}, 1;L^2_x)} \ne  \Vert u_{m^{\prime }}\Vert _{L^1(\frac{1}{2}, 1;L^2_x)}, \ \ \Vert B_m\Vert _{L^1(\frac{1}{2}, 1;L^2_x)} \ne  \Vert B_{m^{\prime }}\Vert _{L^1(\frac{1}{2}, 1;L^2_x)}.\) 

In particular, this yields that
\((u_m, B_m) \ne  (u_{m^{\prime }}, B_{m^{\prime }}),\ \ \forall m\ne  m^{\prime }.\) 
But since zero is not contained in the support of \(\Psi \) ,
by the temporal support result in Theorem REF  \((iii)\) ,
the temporal support of \((u_m, B_m)\)  is contained in \([1/8, 11/8]\) ,
and so
\( u_m(0) = B_m(0) =0, \ \ \forall m\ge 1.\) 
Thus, we obtain infinitely many different weak solutions to (REF )
with the same initial data at time zero.
Concerning the magnetic helicity,
on one hand,
we have from (REF ) that
\( \mathcal {H}_{B_m, B_m} (0) =0, \ \ \forall m\ge 1.\) 
On the other hand, the vector potential \(A_m\)  corresponding to the magnetic field
\(B_m\)  can be computed explicitly by
\(A_m(t,x):= {m\Psi (t)}(\sin x_{3} ,\cos x_{3} , 0 ).\) 

This yields that,
for the magnetic helicity of \((u_m,B_m)\) ,
\(\mathcal {H}_{B_m,B_m}(t)= m^2 {\Psi (t)^2}(2\pi )^3,\) 

and so
\(&\Vert \mathcal {H}_{B_m,B_m}\Vert _{L_t^1(\frac{1}{2},1)}=\frac{m^2}{2 }(2\pi )^3.\) 

Taking into account \({\varepsilon }_*\le 1/100\) 
and the small deviation of the magnetic helicity in Theorem REF  \((v)\) 
we lead to
\( \Vert \mathcal {H}_{B_m,B_m} \Vert _{L_t^1(\frac{1}{2},1)} >0.\) 
Thus, we conclude from (REF ) and (REF ) that
the magnetic helicity \(\mathcal {H}_{B_m, B_m}\)  is not conserved, \(m\ge 1\) .
Therefore, the proof is complete.
\(\square \)

Proof of Theorem  REF
Let \(\left\lbrace \phi _{\varepsilon }\right\rbrace _{\varepsilon >0}\) 
and \(\left\lbrace \varphi _{\varepsilon }\right\rbrace _{\varepsilon >0}\)  be two families of standard compactly support mollifiers
on 3 and \(, respectively.Set{\begin{@align}{1}{-1} u_{n} :=\left(u *_{x} \phi _{\lambda _{n}^{-1}}\right) *_{t} \varphi _{\lambda _{n}^{-1}},\quad B_{n} :=\left(u *_{x} \phi _{\lambda _{n}^{-1}}\right) *_{t} \varphi _{\lambda _{n}^{-1}}.\end{@align}}\) Since \((u,B)\)  is a weak solution to the ideal MHD system,
we infer that there exists a mean-free function \(P_{n}\)  such that
\(\left\lbrace \aligned &\partial _t u_n+\lambda _{n}^{-2\alpha _1}(-\Delta )^{\alpha _1} u_n+ {\mathrm {div}}(u_n\otimes u_n-B_n\otimes B_n)+\nabla P_n={\mathrm {div}}\mathring{R}_n^u , \\&\partial _t B_n+\lambda _{n}^{-2\alpha _2}(-\Delta )^{\alpha _2} B_n+ {\mathrm {div}}(B_n\otimes u_n-u_n\otimes B_n)={\mathrm {div}}\mathring{R}_n^B , \\\endaligned \right.\) 
where the stresses \(\mathring{R}_n^u\)  and \(\mathring{R}_n^B\)  are given by
\(\mathring{R}_n^u:=& \left(u_{n} \mathring{\otimes }u_{n}-B_{n} \mathring{\otimes }B_{n}\right)-\left((u \mathring{\otimes }u-B \mathring{\otimes }B) *_{x} \phi _{\lambda _{n}^{-1}}\right) *_{t} \varphi _{\lambda _{n}^{-1}} \\& +\lambda _{n}^{-2\alpha _1}\mathcal {R}^u (-\Delta )^{\alpha _1} u_{n},  \\\mathring{R}_n^B:=& \left(B_{n} \otimes u_{n}-u_{n} \otimes B_{n}\right)-\left((B \otimes u-u\otimes B) *_{x} \phi _{\lambda _{n}^{-1}}\right) *_{t} \varphi _{\lambda _{n}^{-1}} \\& +\lambda _{n}^{-2\alpha _2}\mathcal {R}^B (-\Delta )^{\alpha _2} B_{n}, \) 

and
\(P_n:= P*_x \phi _{\lambda _n^{-1}} *_t {\varphi }_{\lambda _n^{-1}}-|u_n|^2 + |B_n|^2+ (|u|^2 - |B|^2) *_x \phi _{\lambda _n^{-1}} *_t {\varphi }_{\lambda _n^{-1}}.\) 
Let
\( \nu _1:=\nu _{1,n}:=\lambda _n^{-2\alpha _1},\quad \nu _2:=\nu _{2,n}=\lambda _n^{-2\alpha _2}, \quad \nu _n:=(\nu _{1,n},\nu _{2,n})\) 
and
\(\widetilde{M}:=\max \lbrace  \Vert u \Vert _{H^{\widetilde{\beta }}_{t,x}},\Vert B \Vert _{H^{\widetilde{\beta }}_{t,x}}\rbrace .\) 
Claim: For \(a\)  sufficiently large,
\((u_n,B_n,\mathring{R}_n^u,\mathring{R}_n^B)\)  satisfy the iterative estimates (REF )-() at level \(q=n(\ge 1)\) .
To this end,
let us start with the most delicate estimates in ().
Applying the Minkowski inequality we have
\(&\Vert u -u_{n} \Vert _{L^2_{t,x}} \\\lesssim &\, \left\Vert \int _{ \int _{3}|u(t,x)-u(t-s,x-y)|\phi _{\lambda _{n}^{-1}}(y) \varphi _{\lambda _{n}^{-1}}(s){\rm d}y {\rm d}s_{L^2_{t,x}} \\\lesssim & \int _{ \int _{3} \left\Vert \frac{u(t,x)-u(t-s,x-y)}{(|s|+|y|)^{2+\widetilde{\beta }}} \right\Vert _{L^2_{t,x}}(|s|+|y|)^{2+\widetilde{\beta }} \phi _{\lambda _{n}^{-1}}(y) \varphi _{\lambda _{n}^{-1}}(s){\rm d}y {\rm d}s \\\lesssim & \left\Vert \frac{u(t,x)-u(t-s,x-y)}{(|s|+|y|)^{2+\widetilde{\beta }}} \right\Vert _{L^2_{t,x}L^2_{s,y}}\left\Vert (|s|+|y|)^{2+\widetilde{\beta }} \phi _{\lambda _{n}^{-1}} \varphi _{\lambda _{n}^{-1}} \right\Vert _{L^2_{s,y}}.}Since the Slobodetskii-type norm can be bounded by (see, e.g., \cite [Proposition 1.4]{BO13}){\begin{@align}{1}{-1}\left\Vert \frac{u(t,x)-u(t-s,x-y)}{(|s|+|y|)^{2+\widetilde{\beta }}} \right\Vert _{L^2_{t,x}L^2_{s,y}}\lesssim \Vert u\Vert _{H^{\widetilde{\beta }}_{t,x}}\end{@align}}and{\begin{@align}{1}{-1}\Vert (|s|+|y|)^{2+\widetilde{\beta }} \phi _{\lambda _{n}^{-1}} \varphi _{\lambda _{n}^{-1}} \Vert _{L^2_{s,y}}\lesssim \lambda _n^{- \widetilde{\beta }} \Vert (|s|+|y|)^{2+\widetilde{\beta }} \phi \varphi \Vert _{L^2_{s,y}}\lesssim \lambda _n^{- \widetilde{\beta }},\end{@align}}we obtain that{\begin{@align}{1}{-1} \Vert u-u_{n}\Vert _{L^2_{t,x}}\lesssim &\, \lambda _{n}^{- \widetilde{\beta }} \Vert u\Vert _{H^{\widetilde{\beta }}_{t,x}}\lesssim \lambda _{n}^{- \widetilde{\beta }}\widetilde{M}.\end{@align}}}Moreover,we note that,if \right.\delta _{s,y} u(t,x) :=u(t,x)-u(t-s, x-y),{\begin{@align}{1}{-1} & \left\Vert u_{n} \otimes u_{n}-\left((u \otimes u) *_{x} \phi _{\lambda _{n}^{-1}}\right) *_{t} \varphi _{\lambda _{n}^{-1}} \right\Vert _{L^{1}_{t,x}} \\\lesssim & \Vert  (u-u_{n}) \otimes (u-u_{n}) \Vert _{L^1_{t,x}}+ \bigg \Vert \int _{3}\delta _{s,y} u(t,x) \otimes \delta _{s,y} u(t,x) \phi _{\lambda _{n}^{-1}} \varphi _{\lambda _{n}^{-1}} ds dy \bigg \Vert _{L^1_{t,x}} \\\lesssim & \Vert u-u_n\Vert _{L^2_{t,x}}^2+ \Vert (|s|+|y|)^{4+2\widetilde{\beta }} \phi _{\lambda _{n}^{-1}} \varphi _{\lambda _{n}^{-1}} \Vert _{L^{\infty }_{s,y}}\left\Vert \frac{u(t,x)-u(t-s,x-y)}{(|s|+|y|)^{2+\widetilde{\beta }}} \right\Vert _{L^2_{t,x}L^2_{s,y}}^2 \\\lesssim & \lambda _n^{-2\widetilde{\beta }} \Vert u\Vert _{H^{\widetilde{\beta }}_{t,x}}^2\lesssim \lambda _n^{-2\widetilde{\beta }} \widetilde{M}^2.\end{@align}}Similarly,{\begin{@align}{1}{-1} \left\Vert B_{n} \otimes B_{n}-\left((B \otimes B) *_{x} \phi _{\lambda _{n}^{-1}}\right) *_{t} \varphi _{\lambda _{n}^{-1}} \right\Vert _{L^{1}_{t,x}}\lesssim \lambda _n^{-2\widetilde{\beta }} \widetilde{M}^2.\end{@align}}Estimating as in (\ref {Delta-d-1/2}) and (\ref {e5.18})we have{\begin{@align}{1}{-1} \Vert \lambda _n^{-2{\alpha }_1}\mathcal {R}^u (-\Delta )^{{\alpha }_1} u_n\Vert _{L^1_{t,x}}\lesssim & \lambda _n^{-2{\alpha }_1} \Vert \mathcal {R}^u (-\Delta )^{{\alpha }_1} u_n\Vert _{L^1_{t}L^2_x} \\\lesssim & \lambda _n^{-2{\alpha }_1} \left(\Vert u_n\Vert _{L^1_{t}L^2_x}+ \Vert u_n\Vert _{L^1_{t}L^2_x}^{\frac{3-2{\alpha }_1}{2}} \Vert u_n\Vert _{L^1_{t}H^2_x}^{\frac{2{\alpha }_1-1}{2}} \right) \\\lesssim & \lambda _n^{-2{\alpha }_1} \Vert u_n\Vert _{L^2_{t,x}}+ \lambda _n^{-1} \Vert u_n\Vert _{L^2_{t,x}} \\\lesssim & (\lambda _n^{-2{\alpha }_1} + \lambda _n^{-1}) \widetilde{M}.\end{@align}}\) 
Hence, we conclude from (REF ), (), () and () that
\( \Vert \mathring{R}_n^u\Vert _{L^{1}_{t,x}}& \lesssim \left\Vert u_{n} \otimes u_{n}-\left((u \otimes u) *_{x} \phi _{\lambda _{n}^{-1}}\right) *_{t} \varphi _{\lambda _{n}^{-1}} \right\Vert _{L^{1}_{t,x}} \\&\quad + \left\Vert B_{n} \otimes B_n -\left((B \otimes B) *_{x} \phi _{\lambda _{n}^{-1}}\right) *_{t} \varphi _{\lambda _{n}^{-1}} \right\Vert _{L^{1}_{t,x}} \\&\quad + \Vert \lambda _{n}^{-2\alpha _1}\mathcal {R}^u (-\Delta )^{\alpha _1} u_{n}\Vert _{L^1_{t,x}} \\& \lesssim ( \lambda _n^{-2{\alpha }_1} +\lambda _{n}^{-1}) \widetilde{M}+ \lambda _{n}^{-2\widetilde{\beta }} \widetilde{M}^{2}.\) 

Analogous arguments also yield that
\( \Vert \mathring{R}_n^B \Vert _{L^{1}_{t,x}}\lesssim ( \lambda _n^{-2{\alpha }_2} + \lambda _n^{-1}) \widetilde{M} + \lambda _{n}^{-2\widetilde{\beta }} \widetilde{M}^{2}.\) 
Thus, estimate () is verified at level \(q=n\)  for \(\beta \)  small enough,
where \(\beta >0\)  is as in the proof of Theorem REF .
Moreover, using the Sobolev embedding \(H^3_{t,x} \hookrightarrow L^{\infty }_{t,x}\) 
we have
\( \left\Vert u_{n}\right\Vert _{C_{t, x}^{1}} + \left\Vert B_{n}\right\Vert _{C_{t, x}^{1}}\lesssim & \Vert u_n\Vert _{H^4_{t,x}} + \Vert B_n\Vert _{H^4_{t,x}} \\\lesssim & \lambda _n^4 (\Vert u\Vert _{L^2_{t,x}} + \Vert B\Vert _{L^2_{t,x}})\lesssim \lambda _n^4 \widetilde{M},\) 

which yields () at level \(q=n\) .
Regarding estimate (REF ),
by (REF ) and the Sobolev embedding \(W^{5,1}_{t,x} \hookrightarrow L^{{\infty }}_{t,x}\) ,
\(\Vert \mathring{R}^u_n\Vert _{C^1_{t,x}}\le & \Vert u_{n} \otimes u_{n} - (u \otimes u) *_{x} \phi _{\lambda _{n}^{-1}} *_{t} \varphi _{\lambda _{n}^{-1}} \Vert _{W^{6,1}_{t,x}} \\& + \Vert B_{n} \otimes B_{n} - (B \otimes B) *_{x} \phi _{\lambda _{n}^{-1}} *_{t} \varphi _{\lambda _{n}^{-1}} \Vert _{W^{6,1}_{t,x}}+\lambda _{n}^{-2\alpha _1} \Vert \mathcal {R}^u (-\Delta )^{\alpha _1} u_{n}\Vert _{W^{6,1}_{t,x}} \\=:& J_1 + J_2 + J_3.\) 

Note that
\( J_1\le & \sum \limits _{0\le N_0+ N\le 6} \Vert \partial _t^{N_0}{\nabla }^N(u_{n} \otimes u_{n})\Vert _{L^1_{t,x}}+ \Vert \partial _t^{N_0}{\nabla }^N \left( (u \otimes u) *_{x} \phi _{\lambda _{n}^{-1}} *_{t} \varphi _{\lambda _{n}^{-1}}\right) \Vert _{L^1_{t,x}} \\\lesssim &\sum \limits _{0\le N_{02}+N_{01}+N_1+N_2\le 6}\Vert \partial _t^{N_{01}}{\nabla }^{N_1}u_n\Vert _{L^2_{t,x}} \Vert \partial _t^{N_{02}}{\nabla }^{N_2}u_n\Vert _{L^2_{t,x}} \\& \quad + \sum \limits _{0\le N_0+ N\le 6} \Vert u\Vert _{L^2_{t,x}}^2\Vert {\nabla }^{N} \phi _{\lambda _{n}^{-1}} \Vert _{L^1_t} \Vert \partial _t^{N_0} \varphi _{\lambda _{n}^{-1}} \Vert _{L^1_x} \\\lesssim & \lambda _n^6 \Vert u\Vert _{L^2_{t,x}}^2\lesssim \lambda _n^6 \widetilde{M}^2.\) 

Similarly,
\( J_2 \lesssim \lambda _n^6 \Vert B\Vert _{L^2_{t,x}}^2\lesssim \lambda _n^6 \widetilde{M}^2.\) 

Moreover,
if \({\alpha }_1\in (0, 1/2]\) ,
we have
\( J_3 \lesssim \lambda _n^{-2{\alpha }_1} \Vert u_n\Vert _{H^6_{t,x}}\lesssim \lambda _n^{6-2{\alpha }_1} \Vert u\Vert _{L^2_{t,x}}\lesssim \lambda _n^{6-2{\alpha }_1} \widetilde{M}.\) 

If \({\alpha }_1\in (1/2, 5/4)\) ,
by the interpolation inequality,
\( J_3 \lesssim & \lambda _n^{-2{\alpha }_1} \Vert |{\nabla }|^{2{\alpha }_1-1} u_n\Vert _{H^6_{t,x}}\lesssim \lambda _n^{-2{\alpha }_1} \Vert u_n\Vert _{L^2_{t,x}}^{1-\frac{5+2{\alpha }_1}{8}}\Vert u_n\Vert _{H^8_{t,x}}^{\frac{5+2{\alpha }_1}{8}}\lesssim \lambda _n^5 \widetilde{M}.\) 

Thus, we conclude from (REF )-(REF ) that
\(\Vert \mathring{R}^u_n\Vert _{C^1_{t,x}}\lesssim \lambda _n^6 (\widetilde{M}+ \widetilde{M}^2 ).\) 
Similarly, we have
\( & \Vert \mathring{R}^B_{n} \Vert _{C_{t, x}^{1}}\lesssim \lambda _n^6 (\widetilde{M}+ \widetilde{M}^2).\) 
Thus, taking \(a\)  sufficiently large
we prove estimate (REF ) at level \(q=n\) , as claimed.
Now,
we apply Theorem REF 
to obtain a sequence of solutions \((u_{n,q}, B_{n,q}, \mathring{R}^u_{n,q}, \mathring{R}^B_{n,q})_{q\ge n}\)  satisfying (REF )-().
Letting \(q\rightarrow +\infty \) 
we obtain a weak solution \((u^{(\nu _{n})},B^{(\nu _{n})}) \in H^{\beta ^{\prime }}_{t,x} \times H^{\beta ^{\prime }}_{t,x}\)  to (REF )
for some \(\beta ^{\prime }\in (0,\beta /(7+\beta ))\) .
Moreover,
as in (REF ), it holds that for \(n\ge 1\) ,
\( \left\Vert u^{(\nu _{n})}-u_{n}\right\Vert _{H^{\beta ^{\prime }}_{t,x}}+ \left\Vert B^{(\nu _{n})}-B_{n}\right\Vert _{H^{\beta ^{\prime }}_{t,x}}\le C \sum _{q=n}^{+\infty } \lambda _{q+1}^{-\beta (1-\beta ^{\prime })} \lambda _{q+1}^{7 \beta ^{\prime }} \le \frac{1}{2 n}.\) 
Taking \(\beta ^{\prime }\)  smaller such that
\(0<\beta ^{\prime }<\min \lbrace \widetilde{\beta }, \beta /(7+\beta )\rbrace  \) 
and using () we obtain
\( \Vert u-u_n\Vert _{H^{\beta ^{\prime }}_{t,x}} + \Vert B-B_n\Vert _{H^{\beta ^{\prime }}_{t,x}}\lesssim & \Vert u-u_n\Vert _{L^2_{t,x}}^{1-\frac{\beta ^{\prime }}{\widetilde{\beta }}} \Vert u-u_n\Vert _{H^{\widetilde{\beta }}_{t,x}}^{\frac{\beta ^{\prime }}{\widetilde{\beta }}}+ \Vert B-B_n\Vert _{L^2_{t,x}}^{1-\frac{\beta ^{\prime }}{\widetilde{\beta }}} \Vert B-B_n\Vert _{H^{\widetilde{\beta }}_{t,x}}^{\frac{\beta ^{\prime }}{\widetilde{\beta }}} \\\lesssim & \lambda _n^{-(\widetilde{\beta }- \beta ^{\prime })} \widetilde{M}\le \frac{1}{2n}.\) 
Thus,
we conclude from (REF ) and (REF )
that for \(n\ge 1\) ,
\(& \left\Vert u^{(\nu _{n})}-u\right\Vert _{H^{\beta ^{\prime }}_{t,x}}\le \left\Vert u^{(\nu _{n})}-u_{n}\right\Vert _{H^{\beta ^{\prime }}_{t,x}}+\left\Vert u-u_{n}\right\Vert _{H^{\beta ^{\prime }}_{t,x}} \le \frac{1}{n}, \\& \left\Vert B^{(\nu _{n})}-B\right\Vert _{H^{\beta ^{\prime }}_{t,x}} \le \left\Vert B^{(\nu _{n})}-B_{n}\right\Vert _{H^{\beta ^{\prime }}_{t,x}}+\left\Vert B-B_{n}\right\Vert _{H^{\beta ^{\prime }}_{t,x}} \le \frac{1}{n},\) 

which converge to zero,
thereby yielding (REF ).
Therefore, the proof of Theorem REF  is complete. \(\square \)
Remark 6.1 
We close this section with the remark that,
the weak solutions constructed in [2]} to the ideal MHD equations (REF )
is in the space \(H^{\widetilde{\beta }}_{t,x}\)  for some \(\widetilde{\beta }>0\) .
Though the solutions constructed in [2]} are on the time interval \([0,T]\) ,
with slight modifications the proof in [2]} also applies to the time interval \(.Hence we mainly consider the time interval \)  below.
The constructed solution \((u,B)\in C_tH^{\widetilde{\beta }}_x \times C_tH^{\widetilde{\beta }}_x\)  in [2]},
where \(\widetilde{\beta }>0\) , can be approximated by
the relaxation solutions \((u_q,B_q,\mathring{R}_{q}^u,\mathring{R}_{q}^B)\)  to (REF )
with \(\nu _1=\nu _2=0\) ,
which satisfy the inductive estimates that for any \(q\ge 0\) 
(see [2]})
\(& \Vert {u_q}\Vert _{C_tL_{x}^{2}} \le 1-\delta ^{\frac{1}{2}}_q,\quad \Vert _{C_tL_{x}^{2}} \le 1-\delta ^{\frac{1}{2}}_q, \\& \Vert {u_q}\Vert _{C_{t,x}^{1}} \le \lambda _{q}^{2},\quad \Vert _{C_{t,x}^{1}} \le \lambda _{q}^{2}, \\& \Vert \mathring{R}_q^u\Vert _{C_tL^{1}_{x}} \le c_u \delta _{q+1},\quad \Vert \mathring{R}_q^B\Vert _{C_tL^{1}_{x}} \le c_B \delta _{q+1}, \\& \left\Vert u_{q+1}-u_{q}\right\Vert _{C_tL^{2}_{x}} \le \delta _{q+1}^{\frac{1}{2}}, \quad \left\Vert B_{q+1}-B_{q}\right\Vert _{C_tL^{2}_{x}} \le \delta _{q+1}^{\frac{1}{2}},\) 

where \(\delta _{q+1}=\lambda _{q+1}^{-2\beta }\) , \(\widetilde{\beta }\in (0,\beta /(2+\beta ))\) ,
\(\beta \)  is a small positive constant, and \(c_u, c_B>0\) .
Then, the interpolation inequality yields that
\(&\,\sum _{q \ge 0} \Vert  u_{q+1} - u_q \Vert _{H^{\widetilde{\beta }}_{t,x}} +\sum _{q \ge 0} \Vert  B_{q+1} - B_q \Vert _{H^{\widetilde{\beta }}_{t,x}}\\\le & \, \sum _{q \ge 0} \Vert  u_{q+1} - u_q \Vert _{L^2_{t,x}}^{1- \widetilde{\beta }}\Vert  u_{q+1} - u_q \Vert _{H^1_{t,x}}^{\widetilde{\beta }} +\sum _{q \ge 0} \Vert  B_{q+1} - B_q \Vert _{L^2_{t,x}}^{1- \widetilde{\beta }}\Vert  B_{q+1} - B_q \Vert _{H^1_{t,x}}^{\widetilde{\beta }}\\\le & \, \sum _{q \ge 0} \Vert  u_{q+1} - u_q \Vert _{C_tL^2_{x}}^{1- \widetilde{\beta }}\Vert  u_{q+1} - u_q \Vert _{C^1_{t,x}}^{\widetilde{\beta }} +\sum _{q \ge 0} \Vert  B_{q+1} - B_q \Vert _{C_tL^2_{x}}^{1- \widetilde{\beta }}\Vert  B_{q+1} - B_q \Vert _{C^1_{t,x}}^{\widetilde{\beta }}\\\lesssim &\, \sum _{q \ge 0} \delta _{q+1}^{\frac{1-\widetilde{\beta }}{2}}\lambda _{q+1}^{2 \widetilde{\beta }} <{\infty }, \) 

which yields that \(\lbrace (u_q,B_q)\rbrace _{q\ge 0}\)  also converges in the
Sobolev space \(H^{\widetilde{\beta }}_{t,x} \times H^{\widetilde{\beta }}_{t,x}\) ,
and thus, it follows from the uniqueness of strong limits
that \((u,B)\in H^{\widetilde{\beta }}_{t,x} \times H^{\widetilde{\beta }}_{t,x}\) .

Acknowledgment.
The authors would like to thank Peng Qu for giving the nice lecture on convex integration method
at SJTU this summer and for valuable discussions.
Yachun Li thanks the support by NSFC (No. 11831011),
Deng Zhang thanks the support by NSFC (No. 11871337)
and Shanghai Rising-Star Program 21QA1404500.
This work is also partially supported by Institute of Modern Analysis–A Shanghai Frontier Research Center.
